,antid]ivatives1 ,9tegr,n1 &!
     ,funda;tal ,!orem ( ,calculus

  ,if y rememb] back 6! 2g9n+ (! class1
we sd t calculus 0 ab #2 pro#ms4 ,"o
pro#m 0 f9d+ slopes ( curves2 we h
c]ta9ly resolv$ ? pro#m -pletely4 ,!
second pro#m 0 f9d+ ! >1s "u curves4 ,we
hav5't ?"\ a mo;t ab ? second pro#m s9ce
! f/ week ( class4 ,we "\ 6-e back & see
:at we c d ab x4
  ,weconjectur$ at ! 2g9n+ (! seme/] t !
pro#m ( -put+ slopes &! pro#m ( -put+
>1s seem$ 6be 9v]ses ( "o ano!r4 ,t is1
if g(x) 0 ! slope ( f(x) (:at we h n[
le>n$ 6call f'(x)) !n ! >ea "u ! curve
g(x) seem$ 6be f(x)_4
  ,"! >e a lot ( details 6be "w$ \ 2f we
c ev5 2 -pletely cle> exactly :at claim
is 2+ made "h4 ,b x is cle> t we ne$
6?9k ab -put+ >1s1 & t if we want 6make
progress on \r 3jecture1 !n we %d ?9k ab
! pro#m ( f9d+ f(x) if we >e giv5
f'(x)_4
                                      #j
    ,antid]ivatives4
  ,if f(x) .k ,f'(x), s t ! func;n f(x)
is ! derivative ( ,f(x), !n ! func;n
,f(x) is call$ an antid]ivative ( f(x),
or an 9def9ite 9tegral ( f(x)_4 ,= 9/.e1
if we h
  ,f(x) .k x^3"+sin x+2
  f(x) .k #3x^2"+cos x
!n ;f is ! derivative ( ;,f, & ;,f is an
antid]ivative ( ;f_4 ,x is "r 6say 8an
antid]ivative10 n 8! antid]ivative10
s9ce f(x) has lots ( antid]ivatives1
am;g !m
  ,f(x) .k x^3"+sin x+2
  ,g(x) .k x^3"+sin x-3
  ,h(x) .k x^3"+sin x
  ,k(x) .k x^3"+sin x-17.p^2_4
,cle>ly all ^! func;ns h ! same
derivative1 f(x) .k #3x^2"+cos x_4 ,x
isn't h>d 6prove ! foll[+

  ,!orem3 ,if ,f(x) is any antid]ivative
(a 3t9u\s func;n f(x), !n ,g(x) is an
antid]ivative ( f(x) if & only if
                                      #a
,g(x) .k ,f(x)+c = "s 3/ant ;c_4

  ,pro(3 ,! hypo!sis is t
,f'(x) .k f(x)_4 ,f ? x foll[s immly t
if ,g(x) .k ,f(x)+c, !n
,g'(x) .k ,f'(x) .k f(x), s t ;,g is al
an antid]ivative ( ;f_4
  ,3v]sely1 suppose ,g(x) is an
antid]ivative ( f(x), s t
,g'(x) .k f(x)_4 ,!n ! func;n
,h(x) .k ,g(x)-,f(x) satisfies
  ,h'(x) .k ,g'(x)-,f'(x) .k f(x)-f(x)
    .k #0_4
,if ,h(x) is n a 3/ant1 !n we c f9d #2
po9ts ;a & ;b at : ,h(a) /.k ,h(b)_4 ,!
,m1n ,value ,!orem "!=e says t = "s po9t
;s 2t ;a & ;b,
  ,h'(s) .k ?,h(b)-,h(a)_/b-a# /.k #0,
3tradict+ ! fact t ,h'(x) is "ey": z]o4
,?us1 ,h(x) .k c = "s 3/ant ;c_2 s
,g(x) .k ,f(x)+c, z claim$4

  ,a g5]ic antid]ivative = f(x) is
denot$1 = r1sons t w ev5tu,y 2come cle>1
                                      #b
z
  ,f(x) .k !f(x)dx_4
,?us1 ! example we've be5 look+ at abv
cd 2 writt5 z
  !(3x^2"+cos x)dx .k x^3"+sin x+c,
! +c 9dicat+ t any 3/ant c 2 a4$ to
x^3"+sin x 6produce a valid
antid]ivative4
  ,s9ce antid]ivatives & derivatives
undo "o ano!r1 we h
  !f'(x)dx .k f(x)+c_4
  ?d_/dx#(!f(x)dx) .k f(x)_4

,=mulas = antid]ivatives4
  ,e =mula = derivatives is al a =mula =
antid]ivatives4 ,?us1 ! =mula
  ?d_/dx#(x^3") .k #3x^2
is equival5t to
  !#3x^2"dx .k x^3"+c,
  &
  ?d_/dx#(sin x) .k cos x
  is equival5t to
  !cos xdx .k sin x+c_4
,all \r "w on derivatives has "!=e left
                                      #c
u )a huge collec;n ( 9tegr,n =mulas4 ,h[
c we organize "ey?+ we "k ab
antid]ivatives8
  ,we mi<t 2g9 )! 9tegral equival5t 6!
=mula ?d_/dx#x^n .k nx^n-1_4 ,we cd
write ? z
  !nx^n-1"dx .k x^n"+c,
b a m use;l =mula mi<t 2 replace ;n by
n+1 & !n 6divide by n+1 6get
  !x^n"dx .k ?x^n+1"_/n+1#+c_4
,"o c r,nalize ? 9=m,y 0say+ t
6di6]5tiate x^n, "o decr1ses ! expon5t
;n by #1 & !n multiplies 0! old expon5t
;n_4 ,"!=e 6take an antid]ivative ( x^n,
: is ! rev]se op],n1 "o "\ 69cr1se !
expon5t by #1 & !n divide 0! new expon5t
n+1_4 ,m =m,y1 "o c v]ify ! 9tegral
=mula z "o alw does3 ! 9tegral =mula is
equival5t to
  ?d_/dx#(?x^n+1"_/n+1#) .k x^n,
: c 2 v]ifi$ j 0us+ \r di6]5ti,n =mulas4
  ,? idea is wor? "uscor+4 ,x may n alw
2 easy---& 9de$1 x is n alw possi#---
6f9d a =mula =! antid]ivative (a func;n
                                      #d
f(x)_4 ,b any 3jectur$ 9tegr,n =mula c
easily 0*eck$ 0tak+ ! derivative (!
purport$ 9tegral1 & v]ify+ t xs
derivative is ! orig9al func;n4
  ,"h >e a few o!r basic =mulas =
9def9ite 9tegrals4
  !sin xdx .k -cos x+c
  !cos xdx .k sin x+c
  !#1 dx .k x+c
  !sec^2 xdx .k tan x+c
  !sec xtan xdx .k sec x+c_4
  ,! la/ two ( ^! =mulas >e ! 2g9n+ (a m
g5]al situ,n3 9tegral =mulas (t5 >ise z
a3id5tal 3sequ;es ( calcul,ns 9volv+
derivatives4
  ,x is wor? not+ two g5]al =mulas t let
"o -b9e ^! basic facts & -pute m
antid]ivatives4
  !kf(x)dx .k k!f(x)dx
  !(f(x)+-g(x))dx .k !f(x)dx+-!g(x)dx_4
  ,z usual1 x's easy 6prove ^! 9tegr,n
=mulas 0turn+ !m 96di6]5ti,n =mulas4 ,if
  !f(x)dx .k ,f(x)+c,
!n ,f'(x) .k f(x)_4 ,! =mula
                                      #e
  ?d_/dx#(k,f(x)) .k k,f'(x) .k kf(x)
c !n 2 rewritt5 z
  !kf(x)dx .k k,f(x)+c .k k!f(x)dx,
prov+ ! f/ =mula abv4 ,=! second1
suppose t
  !f(x)dx .k ,f(x)+c
  !g(x)dx .k ,g(x)+c_4
,^! >e equival5t to ,f'(x) .k f(x) &
,g'(x) .k g(x)_4 ,! di6]5ti,n =mula
  ?d_/dx#(,f(x)+-,g(x))
    .k ,f'(x)+-,g'(x) .k f(x)+-g(x)
c !n 2 writt5 z
  !(f(x)+-g(x))dx .k ,f(x)+-,g(x)+c
    .k !f(x)dx+!g(x)dx_4

,?ree h>d] =mulas4
  ,if we're s]i\s 9 say+ t e di6]5ti,n
=mula gives rise 6an 9tegral =mula1 !n
we "\ at l1/ 6look briefly at ! fancie/
di6]5ti,n =mulas we've develop$1 !
,product1 ,quoti5t1 & ,*a9 ,rules1
  ?d_/dx#(f(x)g(x)) .k f'(x)g(x)
    +f(x)g'(x)
  ?d_/dx#(?f(x)_/g(x)#) .k ?g(x)f'(x)
                                      #f
    -f(x)g'(x)_/g(x)^2"#
  ?d_/dx#(f(g(x))) .k f'(g(x))g'(x)_4
,^! 2come 9tegr,n =mulas3
  !(f'(x)g(x)+f(x)g'(x))dx .k f(x)g(x)+c
  !?g(x)f'(x)-f(x)g'(x)_/g(x)^2"#dx .k
    ?f(x)_/g(x)#+c
  !f'(g(x))g'(x)dx .k f(g(x))+c_4
  ,at f/ gl.e1 ^! 9tegr,n =mulas look l
!y wd 2 alm utt]ly use.s = ne>ly e pro#m
( antidi6]5ti,n4 ,surpris+ly (and maybe
depress+ly) !y >e 9/1d \r pr9cipal tools
= 9tegrat+ -plicat$ func;ns4 ,! f/ =mula
gives rise 6a me?od call$ ,9tegr,n
0,"ps1 :ile ! la/ is ! basis (! ev5 m
important te*nique call$ ,9tegr,n
0,sub/itu;n1 or ,*ange ( ,v>ia#4 ,we
won't look at ^! me?ods yet1 b file !
idea away1 & rememb] y saw x "h f/4

,examples4
  ,antid]ivatives ( polynomials "w j z
!y m/4
  !(x^4"-7x^3"+2x-9)dx
    .k ?x^5"_/5#-?7_/4#x^4"+x^2"-9x+c_4
                                      #g
  ,m -plicat$ p[]s ( ;x c 2 tr1t$ ! same
way4
  !(?1_/x^2"#+>x])dx
    .k !(x^-2"+x^1_/2")dx
    .k ?x^-1"_/-1#+,?x^3_/2",_/3_/2,#+c
    .k -?1_/x#+?2_/3#x>x]+c_4
  ,x's wor? not+ t "! is exactly "o p[]
( ;x we _c n[ 9tegrate3 x^-1 .k ?1_/x#_4
,\r g5]al =mula wd give
  !?1_/x#dx .k !x^-1"dx .k ?x^0"_/0#+c,
: c't possibly 2 "r4 ,we'll ne$ 6-e back
6? pro#m d[n ! road4 ,amaz+ly1 ? "o
annoy+ excep;nal case w turn \ 6l1d u
6"s (! mo/ important func;ns 9
ma!matics4
  ,"s"ts seem+ly 9tracta# pro#ms c 2
r5d]$ easy 0"s rewrit+4 ,/&>d examples (
? 9clude ?+s l
  !(?1+x_/>x]#)dx
    .k !(?1_/>x]#+?x_/>x]#)dx
    .k !(x^-1_/2"+x^1_/2")dx
    .k ,?x^1_/2",_/1_/2,#+,?x^3_/2",_/3
      _/2,#+c
    .k #2>x]+?2_/3#x>x]+c_4
                                      #h
  ,"s"ts "o c al play >.d & guess
solu;ns4 ,"h's an example ": "! >e a
c\ple ( approa*es4 ,suppose we want
6-pute
  !sin xcos xdx_4
,we mi<t try 6?9k (a func;n hav+
sin xcos x z xs derivative4 ,a r1sona#
guess mi<t 2 "s?+ l sin^2 x_4 ,if we 7
6try ?1 we cd -pute xs derivative3
  ?d_/dx#sin^2 x .k #2sin xcos x_4
,? isn't q :at we want$1 b x's close4
,:y n divide by #2_8 ,we'd !n h
  ?d_/dx#(?sin^2 x_/2#)
    .k ?2sin xcos x_/2# .k sin xcos x,
s
  !sin xcos xdx .k ?sin^2 x_/2#+c_4
  ,"h's a second ay 6solve ! same pro#m4
,we "k t
  sin (2x) .k #2sin xcos x
  ?1_/2#sin (2x) .k sin xcos x_4
,\r pro#m is "!=e 6-pute
  !?1_/2#sin (2x)dx_4
,we "!=e ne$ 6?9k (a func;n ^:
derivative is ?1_/2#sin (2x)_4 ,a
                                      #i
r1sona# f/ guess mi<t 2 "s?+ l
-?1_/2#cos (2x), & 9de$1 -put+ !
derivative ( ? gives
  ?d_/dx#(-?1_/2#cos (2x))
    .k ?1_/2#sin (2x)*2 .k sin (2x),
: is alm :at we want$4 ,6get x exactly
"r1 let's divide by #2_4 ,we !n h
  ?d_/dx#(-?1_/4#cos (2x))
    .k ?1_/4#sin (2x)*2
    .k ?1_/2#sin (2x),
s t
  !sin xcos xdx .k !?1_/2#sin (2x)dx
    .k -?1_/4#cos (2x)+c_4
  ,wait a m9ute1 ?\<4 ,didn't we j get
#2 di6]5t answ]s = ? 9tegral8 ,no1 b x
takes a bit ( trigonometric cl"e;s 6see
t \r two answ]s >e equival5t4
  -?1_/4#cos (2x)
    .k -?1_/4#(cos^2 x-sin^2 x)
    .k -?1_/4#((1-sin^2 x)-sin^2 x)
    .k ?1_/2#sin^2 x-?1_/4#,
: has ! =m ?sin^2 x_/2#+c_4 ,s \r s5se t
"! is ord] 9 ! cosmos rema9s 9tact4

                                     #aj
,a ,f/ ,di6]5tial ,equ,n4
  ,"! turn \ 6be a va/ numb] ( situ,ns 9
: "o "ks ! derivative (a func;n1 & "o
seeks ! func;n xf4 ,"o /&>d example1 "o
t calculus 0 9v5t$ 6solve1 is ! dynamics
( fall+ bodies4 ,let's />t \ rememb]+
:at we "k ab ? pro#m4
  ,if s(t) is ! posi;n (a mov+ object1
v(t) is xs veloc;y1 & a(t) is xs
a3el],n1 !n v(t) .k s'(t) &
a(t) .k v'(t)_4 ,? m1ns t
  v(t) .k !a(t)dt
  s(t) .k !v(t)dt_4
,if we c take antid]ivatives1 we c "!=e
-pute ! veloc;y & posi;n f ! a3el],n4
  ,? wd 2 use;l1 s9ce we al "k1 f ! "w (
,galileo1 t =a fall+ object ne> ! e>?1 !
a3el],n is a 3/ant1
a(t) ".k<*] -#9.8 ;m_/;s^2_4 ,let's use
! approxim,n a(t) ".k<*] -#10 ;m_/;s^2,
& try 6d a pro#m t ties tgr ne>ly "ey?+
9 ? !ory4
  ,s "h's ! pro#m3

                                     #aa
  ,a p]son /&+ on top (a #40 met] tall
build+ ?r[s a ball up s t x goes up = "o
second 2f r1*+ xs maximum hei<t4 ,x !n
falls pa/ ! />t+ posi;n & ev5tu,y hits !
gr.d4 ,"qs "o cd n[ ask 9clude
  (a) ,:at's a =mula =! veloc;y8
  (;b) ,:at's a =mula =! posi;n8
  (;c) ,:at's ! maximum hei<t8
  (;d) ,:5 does ! ball hit ! gr.d8
  (;e) ,h[ fa/ is x go+ :5 x hits !
gr.d8
  (;f) ,h[ w ! police react 6?8

  ,& "h >e "s solu;ns3
  (a) ,we "k t a(t) .k -#10, & "!=e t
  v(t) .k !a(t)dt .k !-10 dt
    .k -#10t+c_4
,! pro#m is n[ 6f9d a value =! 3/ant
;c_4 ,we cd d ? if "! 7 a "t at : we
knew ! veloc;y4 ,& "! is3 we "k t at
t .k #1, ! veloc;y is #0_4 ,"!=e
  #0 .k v(1) .k -#10(1)+c
  c .k #10_4
,?us1
                                     #ab
  v(t) .k -#10t+10_4
  (;b) ,we d ! same ?+ 6get ! posi;n f !
veloc;y4 ,we "k
  s(t) .k !(-10t+10)dt
    .k -#5t^2"+10t+k_4
,ag1 ! pro#m is 6f9d ! value ( ;k_4 ,we
c d ? 2c we "k t at t .k #0, s .k #40_4
,?us1
  #40 .k s(0) .k -#5(0^2")+10(0)+k .k k,
s t
  s(t) .k -#5t^2"+10t+40_4
  (;c) ,! maximum hei<t o3urs :5
t .k #1, s x is
s(1) .k -#5(1^2")+10(1)+40 .k #45 met]s4
  (;d) ,! ball hits ! gr.d :5
  -#5t^2"+10t+40 .k #0
  t^2"-2t-8 .k #0
  (t-4)(t+2) .k #0
  t .k #4 or t .k -#2_4
,obvi\sly t .k -#2 doesn't make physical
s5se1 s ! ball m/ hit ! gr.d :5
t .k #4_4
  (;e) ,:5 t .k #4, ! veloc;y is
v(4) .k -#10(4)+10 .k -#30 met]s p]
                                     #ac
second4
  (;f) ,probably badly4 ,x cd al 2 "s
attorney's ticket 6a major home
improve;t project4

    ,>1s & ,riemann ,sums

,3ceptual me?od4
  ,if y ?9k back on ! 2g9n+s ( \r "w )
derivatives1 y c describe \r me?od =
-put+ ! slopes ( tang5t l9es z a process
( #3 /eps3
  (1) ,give up4 ,y c't -pute slopes4
  (2) ,try 9/1d 6approximate ! slope
0tak+ ! slope (! secant l9e jo9+ two
close tgr po9ts4
  (3) ,n[ make ! approxim,n 96an exact
result 0tak+ ! limit z ! 4t.e 2t ! two
po9ts goes to #0_4

  ,we n[ want 6/>t "w on ! second major
pro#m (! calculus1 ! -put,n (! >ea "u an
>bitr>y curve4 ,s9ce we su3e$$ s well )!
slope pro#m1 x seems r1sona# 6try 6use !
                                     #ad
same approa* 6-pute ! >ea "u a curve4 ,!
?ree-fold me?od wd !n say1
  (1) ,give up4 ,y c't -pute >1s4
  (2) ,try 9/1d 6approximate ! >ea
0br1k+ ! region 96a bun* ( slices &
approximat+ ! >ea 9 ea* slice 0! >ea (a
rectangle ^: hei<t is ! hei<t (! func;n
at "s po9t 9 ! slice4 ,add up ! >1s (
all ^! rectangles4
  (3) ,n[ make ! approxim,n 96an exact
result 0tak+ ! limit z ! numb] ( slices
approa*es ,= &! wid? ( ea* 9dividual
slice goes to #0_4

  ,figure #1 %[s a typical curve &!
rectangles we mi<t use 9 ^? slices4

,=mulas4
  ,suppose we want 6try ? me?od 6-pute !
>ea "u ! graph ( y .k f(x) 2t x .k a &
x .k b_4 ,we mi<t 2g9 0slic+ ! 9t]val
@(a, b@) 9to ;n slices ( equal wid?
?b-a_/n#_4 ,! po9ts divid+ ! slices wd
!n 2 at
                                     #ae
  a+0(?b-a_/n#), a+1(?b-a_/n#),
    a+2(?b-a_/n#), a+3(?b-a_/n#), ''',
    a+n(?b-a_/n#)_4
  ,notice t ! fur/ ( ^! po9ts is a, &!
la/ is ;b_4 ,if we build rectangles 9
ea* slice 0us+ ! hei<t (! func;n ;f at !
"r h& 5d (! slice1 !n ! f/ rectangle has
wid? ?b-a_/n#, x has hei<t
  f(a+1(?b-a_/n#)),
& "!=e x has >ea
  f(a+1(?b-a_/n#))(?b-a_/n#)_4
,! second rectangle has wid? ?b-a_/n#,
hei<t
  f(a+2(?b-a_/n#)),
& >ea
  f(a+2(?b-a_/n#))(?b-a_/n#),
& s on4 ,! sum (! >1s ( all ! rectangles
is
  ;,a
    .k f(a+1(?b-a_/n#))(?b-a_/n#)+f(a+2(
      ?b-a_/n#))(?b-a_/n#)+ '''
    +f(a+n(?b-a_/n#))(?b-a_/n#)
    .k f(a+1(?b-a_/n#))+f(a+2(?b-a_/n#))
      + '''
                                     #af
    +f(a+n(?b-a_/n#))(?b-a_/n#)_4
,"o cd al write ? 9 ,sigma not,n z
  ;,a
    .k ".,s%k .k #1<n]@(f(a+k(?b-a
      _/n#))(?b-a_/n#)@)
    .k (?b-a_/n#)".,s%k .k #1<n]@(f(a+k(
      ?b-a_/n#))@)_4
,! exact >ea w 2 ! limit ( ? sum z
n $o ,=_4
  ,a sket* ( ? situ,n is %[n z ,figure
#2_4 ,9 ! ,figure1 we use ! %orth& .,dx
6denote ?b-a_/n#_4

,example #1_3 a /rai<t l9e4
  ,? approa* seems 6make s5se1 b x wd
c]ta9ly help 6see "s examples4 ,let's
take ! simple/ case we cd1 ! >ea "u !
/rai<t l9e y .k #3x 2t x .k #0 &
x .k #1_4 ,we "k ! answ]1 s9ce ! >ea is
j ! >ea (a triangle2 b let's make sure
\r g5]al me?od "ws 9 ? case4 ,if we cut
! 9t]val @(0, 1@) 9to ;n equal wid?
slices1 !n ! cuts wd 2 at ! po9ts
  #0, ?1_/n#, ?2_/n#, ?3_/n#, ''',
                                     #ag
    ?n_/n# .k #1_4
,! sum (! >1s (! rectangles wd "!=e 2
  ;,a
    .k f(?1_/n#)?1_/n#+f(?2_/n#)?1_/n#
      +f(?3_/n#)?1_/n#+ ''' +f(?n_/n#)?1
      _/n#
    .k .(f(?1_/n#)+f(?2_/n#)+f(?3_/n#)
      + ''' +f(?n_/n#).)?1_/n#
    .k .(?3_/n#+?6_/n#+?9_/n#+ ''' +?3n
      _/n#.)?1_/n#
    .k .(1+2+3+ ''' +n.)?3_/n^2"#_4
,n[ if we want 6"w \ ? sum1 we h
6rememb] h[ 6evaluate
  ,s .k #1+2+3+ ''' +(n-2)+(n-1)+n_4
,"! >e lots ( ways 6d ?4 ,"o trick is
6write ! sum 9 two di6]5t ways3
  ,s .k #1+2+3+ ''' +(n-2)+(n-1)+n
  ,s .k n+(n-1)+(n-2)+ ''' +3+2+1_4
,if y add bo? ^! expres.ns 9 columns1 y
get
  #2,s
    .k (n+1)+(n+1)+(n+1)+ ''' +(n+1)+(n+
      1)+(n+1)
    .k n(n+1),
                                     #ah
f : x foll[s t
  ,s .k #1+2+3+ ''' +(n-2)+(n-1)+n
    .k ?n(n+1)_/2#_4
  ,>m$ ) ? fact1 we c n[ -e back 6! >ea
we 7 try+ 6-pute4 ,we h
  ;,a
    .k .(1+2+3+ ''' +n.)?3_/n^2"#
    .k ?n(n+1)_/2#*?3_/n^2"#
    .k ?3(n+1)_/2n# .k ?3n+3_/2n#_4
  ,? 0 \r approxim,n 6! >ea gott5 0a4+
up ! >1s (a bun* ( rectangles4 ,! exact
>ea %d 2 ! limit ( ? expres.n z ! numb]
( slices approa*es ,=, s9ce 9 v n>r[
slices1 ! hei<t (! rectangles &! hei<t (
y .k f(x) >e v ne>ly ! same4 ,9 o!r ^ws1
! exact >ea %d 2
  "lim%n $o ,=] ?3n+3_/2n#
    .k "lim%n $o ,=] ,??3n_/n#+?3_/n#,_/
      ?2n_/n#,#
    .k "lim%n $o ,=] ,?3+?3_/n#,_/2,#
      .k ?3_/2#_4
,? is j :at ! =mula ,a .k ?1_/2#bh =!
>ea (a triangle wd h giv5 u1 s \r me?od
has deliv]$ ! "r answ]4
                                     #ai
  ,:at we've j d"o is 6help 6sub/antiate
\r me?od = -put+ >1s1 0%[+ t x gives !
correct answ] 9 "o (! few cases ": we
alr "k ! >ea4 ,9 s do+1 we've al -put$ !
>ea (a triangle1 n 9 ! mo/ -plicat$
possi# way "kn1 b 9 "o t -es pretty
close4

,example #2_3 a p>abola4
  ,let's try a second example1 "o ": we
c't -pute ! >ea directly4 ,suppose we
want ! >ea "u ! p>abola y .k t^2 2t
t .k #0 & t .k x_4 ,t is1 we want a
g5]ic "r h& 5dpo9t1 x, = \r 9t]val4
  ,if we cut ! 9t]val @(0, x@) 9to ;n
equal wid? slices1 !n ! cuts wd 2 at !
po9ts
  #0, ?1x_/n#, ?2x_/n#, ?3x_/n#, ''',
    ?nx_/n# .k x_4
,! sum (! >1s (! rectangles wd "!=e 2
  ;,a
    .k f(?1x_/n#)?x_/n#+f(?2x_/n#)?x_/n#
      +f(?3x_/n#)?x_/n#+ ''' +f(?nx_/n#)
      ?x_/n#
                                     #bj
    .k .(f(?1x_/n#)+f(?2x_/n#)+f(?3x
      _/n#)+ ''' +f(?nx_/n#).)?x_/n#
    .k .(?1^2"x^2"_/n^2"#+?2^2"x^2"
      _/n^2"#+?3^2"x^2"_/n^2"#+ ''' +
      ?n^2"x^2"_/n^2"#.)?x_/n#
    .k .(1^2"+2^2"+3^2"+ ''' +n^2".)
      ?x^3"_/n^3"#_4
,n[ if we want 6"w \ ? sum1 we h
6rememb] h[ 6evaluate
  ,s .k #1^2"+2^2"+3^2"+ ''' +n^2_4
,? "o isn't z simple z ! previ\s "o4 ,y
probably n"e saw ! =mula 9 s*ool4 ,x
turns \ "! is a =mula1 ?\<1 : is
  ,s .k #1^2"+2^2"+3^2"+ ''' +n^2
    .k ?n(n+1)(2n+1)_/6#_4
,x's n too h>d 6prove ? 09duc;n1 b ,i'd
r n take ! "t 6d s "r n[4 ,let's j
assume ! =mula is "r1 & see ": we 5d up4
  ,! >ea we 7 try+ 6-pute is
  ;,a
    .k .(1^2"+2^2"+3^2"+ ''' +n^2".)
      ?x^3"_/n^3"#
    .k ?n(n+1)(2n+1)_/6#*?x^3"_/n^3"#
    .k ?2n^2"+3n+1_/6n^2"#x^3_4
                                     #ba
  ,? 0 \r approxim,n 6! >ea gott5 0a4+
up ! >1s (a bun* ( rectangles4 ,! exact
>ea %d 2 ! limit ( ? expres.n z ! numb]
( slices approa*es ,=, s9ce 9 v n>r[
slices1 ! hei<t (! rectangles &! hei<t (
y .k f(x) >e v ne>ly ! same4 ,9 o!r ^ws1
! exact >ea %d 2
  "lim%n $o ,=] ?2n^2"+3n+1_/6n^2"#x^3
    .k ?2_/6#x^3 .k ?x^3"_/3#_4
(,rememb] t if ! num]ator & denom9ator
(a r,nal func;n h equal degrees1 !n !
limit (! func;n at ,= is ! ratio (! l1d+
coe6ici5ts_4)

  ,is ? gd or bad news8 ,on ! b"r side1
we got an >ea1 ! f/ re,y new >ea we've
be5 a# 6-pute4 ,ev5 bett]1 :at we've d"o
"h is 6/>t )! func;n f(t) .k t^2 &
6-pute ! >ea func;n
  ,f0(x) .k ?x^3"_/3#_4
,\r 3jecture f ! lab 0 t
,f'0(x) .k f(x), & t 3jecture "ws
p]fectly "h3
  ,f'0(x) .k ?d_/dx#(?x^3"_/3#) .k x^2
                                     #bb
    .k f(x)_4
  ,"! is a bad side 6? calcul,n1 ?\<3 x
wasn't v simple4 ,9 fact1 we 7 only a#
6d x at all 2c (! =mula ,i pull$ \ (!
hat =
  ".,s%k .k #1<n]k^2_4
,x doesn't look l f(x) wd h 6get m* m
-plicat$ 2f we cdn't c>ry \ ! algebra at
all4 ,we "!=e seem 6h a !oretic,y
su3ess;l me?od t may 9 practice 2 too
-plicat$ 6use4


,! ,riemann ,9tegral4
  ,x doesn't look l we c (t5 evaluate !
sums t \r te*niques = -put+ >1s
produces4 ,x al turns \ t = te*nical
r1sons1 us+ ! "r h& 5d (! 9t]vals isn't
alw ! be/ ?+ 6d4 ,) ^! ?"\s 9 m9d1
,riemann us$ \r idea 6write a =mal
def9i;n =! >ea "u ! graph ( f(x) 2t
x .k a & x .k b, a "k b_4
  ,let
a .k x0 "k x1 "k x2 "k ''' "k x;n .k b_4
                                     #bc
,s* a divi.n (! 9t]val @(a, b@) 9to ;n
slices ( possibly di6]5t l5g?s is call$
a "pi;n ( @(a, b@)_4 ,denote ! "pi;n z
;,p_4
  ,= e ;k, #1 "k: k "k: n, let
  .,dx;k .k x;k"-x;k-1
2 ! wid? ( 9t]val ;k 9 ? "pi;n4 ,al1 = e
;k, let x;k-1 "k: c;k "k: x;k, s t c;k
is a po9t 9 slice ;k (! "pi;n4 ,an
approxim,n 6! >ea "u ;f wd !n 2
  f(c1).,dx1+f(c2).,dx2+f(c3).,dx3+ '''
    +f(c;n").,dx;n
    .k ".,s%k .k #1<n]f(c;k").,dx;k_4
,? sum is call$ a ,riemann sum4 ,figure
#3 %[s a picture ( "o ,riemann sum4
  ,a ,riemann sum is an approxim,n 6!
exact >ea "u ! curve1 gott5 0a4+ up !
>1s ( "s rectangles4 ,! exact >ea "u !
curve %d 2 a limit (! ,riemann sums z
all ! 9t]vals 9 ! "pi;n ;,p get small4
,"o way 6say ? precisely is 6let
\\,p\\ .k max .(.,dx;k_3#1 "k: k "k: n.)
2 ! l>ge/ (! slices 9 ! "pi;n ;,p_4 ,!n
! exact >ea "u ;f %d 2
                                     #bd
  "lim%\\,p\\ $o #0] ".,s%k .k #1<n]f
    (c;k").,dx;k,
if ? limit exi/s4 ,? limit is call$ !
,riemann 9tegral ( ;f, & is denot$
  "lim%\\,p\\ $o #0] ".,s%k .k #1<n]f
    (c;k").,dx;k .k !;a^b"f(x)dx_4
,calculus books (t5 d a few ?+s )
,riemann 9tegrals 9 g5]al1 b we won't d
m* ) !m4 ,x is p]fectly r1sona# 6?9k (
!m z j sli<t v>i,ns on \r e>li] sums4
,af all1 \r e>li] sums 7 j special cases
(! sums abv 9 : e .,dx;k .k ?b-a_/n# & 9
: e c;k .k x;k_4 ,= simple 3t9u\s
func;ns1 "! is no?+ 6be ga9$ 0"w+ )
g5]al ,riemann sums1 ?\< !y >e nec if y
want 6f9d (or ev51 to def9e) ! >ea "u
hi<ly 4cont9u\s func;ns4

,a not,nal 9t]lude4
  ,! not,n us$ =! ,riemann 9tegral looks
odd1 j z ! not,n =! antid]ivative look$
odd4 ,if y ?9k ab x a bit1 ?\<1 x 2comes
wond];l4

                                     #be
  ,leibniz,8 not,n =! derivative 0
?dy_/dx# 2c ,leibniz ?"\ (! derivative z
re,y alm ! slope ?.,dy_/.,dx# (a secant
l9e4 ,9 ,leibniz0' not,n1 6move f !
slope (! secant l9e 6! slope (! tang5t
l9e1 t is1 6take a limit1 y j *ange !
,greek .,d 96a ,lat9 ;d_4
  ,! same ?+ "ws ) >1s4 ,an approximate
>ea looks l
  ".,s%k .k #1<n]f(c;k").,dx;k_4
,6get an exact >ea1 y "\ 6turn ! ,greek
lrs 96,lat9 lrs4 ,t is1 ! .,d %d 2come a
;d, &! .,s %d 2come an ;,s_4 ,! sum ran
ov] po9ts x;k .k x & c;k .k x 2t ;a &
;b_4 ,9 ! limit1 we "\ 6be look+ at a
bun* ( po9ts spr1d ov] ? 9t]val4 ,s we
%d write "s?+ l
  ,s;a^b"f(x)dx_4
,/ret* \ ! ;,s 6make x look m grace;l1 &
y h
  !;a^b"f(x)dx_4
,s = >1s z = slopes1 y denote ! limit 0j
a *ange ( alphabets4 ,! 9tegral sign is
! residue (! sum1 &! dx is ! residue (!
                                     #bf
.,dx af y take ! limit4
  ,?9k+ ( 9tegrals r\<ly z 9f9ite sums
is n q precise1 b x is a grt way 6ga9
9si<ts & 9tui;ns ab !m4

,basic ,=mulas4
  ,! two limits ( sums we pa9;lly -put$
abv tell u t
  !;0^1"3xdx .k ?3_/2#,
& t
  !;0^x"t^2"dt .k ?x^3"_/3#_4
  ,gett+ m =mulas ( ? sort seems 6be a
lot ( "w4 ,>e "! g5]al =mulas we c get
0j ?9k+ 3ceptu,y1 ?\<8 ,yes1 "h >e a
few3
  !;a^a"f(x)dx .k #0_4
,? j says t ! >ea "u y .k f(x) 2t x .k a
& x .k a is #0_4 ,? is c]ta9ly s5si#1
s9ce ! >ea is a rectangle ( wid? #0.
  !;a^b"f(x)dx+!;b^c"f(x)dx
    .k !;a^c"f(x)dx_4
,? j says t if y take ! >ea "u ;f 2t
x .k a & x .k b, & y add ! >ea "u ;f 2t
x .k b & x .k c, y get ! >ea "u ;f 2t
                                     #bg
x .k a & x .k c, : surely makes s5se4
  ,9 mak+ ? >gu;t1 we've assum$
implicitly t a "k b "k c_4 ,:at if ?
fails 6be true8 ,well1 \r desire 6make ?
=mula alw "w wd =ce u 6say
  !;a^b"f(x)dx+!;b^a"f(x)dx .k !;a^a
    "f(x)dx .k #0
  !;b^a"f(x)dx .k -!;a^b"f(x)dx_4
,? seems l an e35tric *oice =a def9i;n1
b if we want \r =mulas s f> 6"w )\t
re/ric;n1 !n we >e =c$ 6a3ept ? def9i;n4
,9 ma!matics1 ! def9i;ns (t5 foll[ !
!orems1 & >e design$ 6make ! !orems /ay
true2 s x is n surpris+ t ! -mun;y has
settl$ on ! agree;t t >ea -put$ f "r
6left is ! negative ( >ea -put$ f left
6"r4
  ,= any 3/ant ;,m,
  !;a^b",mf(x)dx .k ,m!;a^b"f(x)dx_4
,"o cd def5d ? geometric,y 0>gu+ t ! >ea
"u y .k ,mf(x) m/ surely 2 ;,m "ts ! >ea
2n y .k f(x)_4 ,af all1 y'd approximate
! >ea 0a4+ up a bun* ( rectangles1 &
multiply+ ! hei<t (a rectangle by ;,m
                                     #bh
multiplies xs >ea by ;,m z well4 ,"o cd
al use ,riemann's def9i;n 6say t
  !;a^b",mf(x)dx
    .k "lim%\\,p\\ $o #0]
      ".,s%k .k #1<n],mf(c;k").,dx;k
    .k "lim%\\,p\\ $o #0] ,m
      ".,s%k .k #1<n]f(c;k").,dx;k
    .k ,m"lim%\\,p\\ $o #0]
      ".,s%k .k #1<n]f(c;k").,dx;k
    .k ,m!;a^b"f(x)dx_4
,! f/ & la/ equalities "h >e ! def9i;n
(! ,riemann 9tegral2 ! second is !
4tributive law2 ! ?ird is ! fact t
3/ants c 2 pull$ \ ( limits4
  ,"o t's m obvi\s algebraic,y ?an x is
geometric,y is
  !;a^b"(f(x)+g(x))dx
    .k !;a^b"f(x)dx+!;a^b"g(x)dx_4
,we cd get ? f ! def9i;n (! ,riemann
9tegral 0writ+
  !;a^b"(f(x)+g(x))dx
    .k "lim%\\,p\\ $o #0]
      ".,s%k .k #1<n](f(c;k")+g(c;k
      ")).,dx;k
                                     #bi
    .k "lim%\\,p\\ $o #0]
      ".,s%k .k #1<n](f(c;k").,dx;k"
      +g(c;k").,dx;k")
    .k "lim%\\,p\\ $o #0]
      ".,s%k .k #1<n]f(c;k").,dx;k"+
      "lim%\\,p\\ $o #0] ".,s%k .k #1<n]
      g(c;k").,dx;k
    .k !;a^b"f(x)dx+!;a^b"g(x)dx_4
  ,6?9k ab ? geometric,y takes a mo;t1
?\<4 ,suppose f(x) & g(x) >e ! func;ns
%[n 9 ,figures #4 & #5, resp4 ,!n !
=mula says t ! >ea "u y .k f(x)+g(x),
%[n 9 ,figure #6, %d 2 ! sum (! >1s ( ^!
#2 regions4 ,:y %d t 28
  ,"o way 6/>t 96? "q is 6look at !
graphs ( f(x) & f(x)+g(x) tgr1 z 9
,figure #7_4 ,we'd 2 d"o if we cd "u/&
:y ! top region 9 ? ,figure #7 has !
same >ea z ! >ea 9 ,figure #5_4
  ,algebraic,y1 ? is true 2c if we 7
6cut bo? regions 96slices & approximate
! >1s ) rectangles1 9 bo? cases !
rectangle ne> ;x wd h hei<t ab g(x)_4
,if all ! rectangles h ! same size1 !n !
                                     #cj
total >1s h 6h ! same size z well1 &
we're d"o4
  ,? obs]v,n is a special case (a g5]al
pr9ciple call$ ,cavali]i's ,pr9ciple1 af
an ,italian ma!matician (! g5],n 2f
,newton4 ,cavali]i ass]t$ t if y h two
regions1 ;,a & ;,b, & if e v]tical l9e
9t]sects ;,a 9 a seg;t exactly z l;g z
xs 9t]sec;n ) ;,b, !n ;,a & ;,b h ! same
>ea4 ,8 seg;ts play ! same role z \r
rectangles4 ,cavali]i's >gu;t = ? 0
deli<t;l3 ,imag9e1 he sd1 t ea* figure 0
drawn on clo?4 ,!n = e ?r1d1 ! l5g? ( t
?r1d t meets ;,a equals ! l5g? ( t ?r1d
t meets ;,b_4 ,figures ;,a & ;,b m/ "!=e
h ! same >ea1 s9ce !y require ! same
am.t ( fabric 6make4

    ,! ,funda;tal ,!orem ( ,calculus
  ,2f ,i />t on ! ,funda;tal ,!orem (
,calculus1 let me make a su7es;n3 ,9
a4i;n 6^! notes1 x is v m* wor? r1d+ yr
text or o!r tr1t;ts (! subject c>e;lly4
,:at we're do+ n[ is ! 3ceptual c5t] (
                                     #ca
calculus4 ,x's h>d & subtle1 & x's a gd
idea 6look at x m ?an once1 f di6]5t
p]spectives4
  ,! ,funda;tal ,!orem ( ,calculus is !
!orem t tells u t ! pro#m ( f9d+ slopes
&! pro#m ( f9d+ >1s >e 9v]ses ( ea* o!r1
"!by lett+ u f9,y make precise & prove a
3jecture we've _h s9ce ! f/ "d (! class4
,! ,funda;tal ,!orem lets u use
antid]ivatives 6-pute >1s1 & x unites !
two di6]5t m1n+s we h =! 9tegral4

,! ,>ea func;n & xs derivative4
  ,9 ! lab1 we look$ at ! func;n
  ,f;a"(x) .k !;a^x"f(t)dt,
! >ea "u ! graph ( y .k f(t) 2t t .k a &
t .k x_4 ,\r 3jecture1 = : we've be5
ga!r+ /r;g] & /r;g] evid;e1 is t
  ,f';a"(x) .k ?d_/dx#(!;a^x"f(t)dt)
    .k f(x)_4
,? is "o =m (v>i\sly call$ ! f/ =m or !
second =m1 dep5d+ on :at c\rse ( logical
develop;t y're tak+) (a result prop]ly
call$ ! ,funda;tal ,!orem ( ,calculus4
                                     #cb
  ,calculus books (t5 prove ? !orem
"s:at m precisely ?an ,i plan to1 & f a
sli<tly di6]5t p]spective4 ,let me try
6write precisely ! >gu;t = xs tru? t ,i
hope y 7 l$ 64cov] 9 ! lab4
  ,6-pute ,f';a"(x), we ne$ 6f9d
  "lim%h $o #0] ?,f;a"(x+h)
    -,f;a"(x)_/h#_4
,! num]ator ( ? di6];e quoti5t is !
di6];e ( two numb]s4 ,f;a"(x+h) is ! >ea
"u ! graph ( y .k f(t) 2t t .k a &
t .k x+h_4 ,simil>ly1 ,f;a"(x) is ! >ea
"u ! graph ( y .k f(t) 2t t .k a &
t .k x_4 ,! di6];e1 ,f;a"(x+h)-,f;a"(x)
is "!=e ! >ea "u ! graph ( y .k f(t) 2t
t .k x & t .k x+h_4 ,? is %[n pictori,y
2l 9 ,figures #8 & #9_4
  ,:5 ;h is small1 ! di6];e
,f;a"(x+h)-,f;a"(x) "!=e repres5ts an
>ea : is v ne>ly a trapezoid ^: wid? is
;h & ^: hei<ts >e f(x) & f(x+h)_4 ,!
trapezoid >ea =mula "!=e tells u t v
ne>ly1
  ,f;a"(x+h)-,f;a"(x)
                                     #cc
    .k h*(?f(x)+f(x+h)_/2#)_4
,? approxim,n %d get bett] & bett] z
h $o #0, s9ce !n ! v>i,n 9 hei<t acr !
trapezoid %d 2 less & less4 ,z a result1
x seems t we %d h
  ,f';a"(x)
    .k "lim%h $o #0] ?,f;a"(x+h)-,f;a
      "(x)_/h#
    .k "lim%h $o #0] @(?1_/h#(,f;a"(x+h)
      -,f;a"(x))@)
    .k "lim%h $o #0] @(?1_/h#(h*(?f(x)
      +f(x+h)_/2#))@)
    .k "lim%h $o #0] (?f(x)+f(x+h)_/2#)
    .k ?f(x)+f(x)_/2# .k f(x)_4
,? proves ! 3jecture f ! lab1 &) x1 ? =m
(! ,funda;tal ,!orem ( ,calculus4
  ,"! >e a c\ple ( po9ts at : "! has be5
a bit ( frly h& wav+ "h4 ,9 -put+ !
limit abv1 we've cle>ly assum$ t ;f is a
3t9u\s func;n4 ,we've al assum$ r
vaguely t ;f doesn't *ange too wildly 2t
;x & x+h, s t ! trapezoid is a r1sona#
approxim,n 6! >ea4 ,? c all 2 ju/ifi$
precisely us+ ! ,m1n ,value ,!orem1 b
                                     #cd
6me1 x seems cle>ly s5si#4
  ,"! is a ano!r =m (! ,funda;tal ,!orem
( ,calculus1 (the second or f/ =m1
dep5d+ on :om y li/5 to)1 : says !
foll[+3

  ,if ,f(x) is an antid]ivative ( f(x),
s t ,f'(x) .k f(x), !n
  !;a^b"f(t)dt .k ,f(b)-,f(a)_4
  ,? result turns \ 6be a coroll>y ( \r
e>li] =m (! ,funda;tal ,!orem4 ,we "k t
if
  ,f;a"(x) .k !;a^x"f(t)dt,
!n ,f';a"(x) .k f(x)_4 ,?us1 bo?
,f;a"(x) & ,f(x) >e antid]ivatives (
f(x)_4 ,"! is "!=e "s 3/ant ;c s* t
  ,f;a"(x) .k ,f(x)+c_4
,all we ne$ 6d is 6f9d ! value ( ;c_4
,6d ?1 we ne$ 6look = "s value ( ;x at :
we "k ,f;a"(x)_4 ,! natural *oice is
x .k a_4 ,f;a"(a) is ! >ea "u ! graph (
f(t) 2t t .k a & t .k a_4 ,9 o!r ^ws1
  ,f;a"(a) .k #0_4
,plu7+ ? 96! e>li] equ,n1 we get
                                     #ce
  #0 .k ,f;a"(a) .k ,f(a)+c
  c .k -,f(a)_4
,?us1 = any ;x, we h
  !;a^x"f(t)dt .k ,f;a"(x) .k ,f(x)+c
    .k ,f(x)-,f(a)_4
,plu7+ 9 x .k b gives ! !orem4

,a brief celebr,n4
  ,? is a re,y cool result6 ,n only h we
be5 want+ 6prove ? all seme/]1 b x's v
p[];l4 ,x m1ns t we c n[ -pute ! >ea "u
! graph ( any func;n ^: antid]ivative we
c f9d4 ,t's n q e func;n1 nor w x "e 22
b x is q a lot ( func;ns4

,examples4
  ,we _h 6"w h>d ! o!r "d 6-pute ! >ea
"u ! graph ( y .k t^2 2t t .k #0 &
t .k x_4 ,n[ x's easy3 ,an antid]ivative
( f(t) .k t^2 is ,f(t) .k t^3"_/3, s
  !;0^x"t^2"dt .k ,f(x)-,f(0)
    .k ?x^3"_/3#-?0^3"_/3#
    .k ?x^3"_/3#_4

                                     #cf
  ,x's equ,y easy 6f9d ! >ea "u ! graph
( y .k t^n 2t t .k #0 & t .k x_4 ,an
antid]ivative ( f(t) .k t^n is
,f(t) .k ?t^n+1"_/n+1#, s
  !;0^x"t^n"dt .k ,f(x)-,f(0)
    .k ?x^n+1"_/n+1#-?0^n+1"_/n+1#
    .k ?x^n+1"_/n+1#_4
,two "ds ago1 "! 7 only two values ( ;n
(0 & #1) at : we knew ! value (
  !;0^x"t^n"dt_4
,a mo;t ago1 "! 7 only #3 s* values (0,
#1, & #2)_4 ,n[ "! >e 9f9itely _m1 & x
wasn't ev5 h>d4 ,we're mak+ fanta/ic
progress4
  ,"h's ano!r example ( \r pres5t
pr[ess3 ,:at's ! >ea "u #1 hump (!
func;n y .k sin x_8 ,t is1 :at's
  !;0^.p"sin xdx=
  ,an antid]ivative ( f(x) .k sin x is
,f(x) .k -cos x, s
  !;0^.p"sin xdx .k (-cos .p)-(-cos #0)
    .k -(-1)-(-1) .k #2_4
,x doesn't get m* simpl] ?an t4

                                     #cg
,"o bit ( not,n4
  ,9 ord] 6cut d[n writ+1 p (t5 write
,f(b)-,f(a) z ,f(x)\;a^b, or z
@(,f(x)@);a^b_4 ,? way1 "o c -pute >1s l
?3
  !;1^3"(x^4"-11x^3"+x^2"-5)dx
    .k @(?x^5"_/5#-?11x^4"_/4#+?x^3"_/3#
      -5x@);1^3
    .k @(?3^5"_/5#-?11(3^4")_/4#+?3^3"_/
      3#-5(3)@)
    -@(?1^5"_/5#-?11(1^4")_/4#+?1^3"_/3#
      -5(1)@)
    .k -?3306_/20#-(-?433_/60#) .k -?
      2494_/15#_4

,,ftc relates ! #2 types ( 9tegral4
  ,let's try 6?9k =a bit & reflect
3ceptu,y on ! m1n+ ( :at we've n[ prov54
  ,"o way 6"u/& ! ,funda;tal ,!orem (
,calculus is 6say t x ties tgr ! two
m1n+s we h =! 9tegral sign4 ,let u
rememb] :at ^! >e4
  ,! def9ite 9tegral1
  !;a^b"f(t)dt,
                                     #ch
is a numb]1 denot+ ! >ea "u ! graph ( ;f
2t t .k a & t .k b_4
  ,! 9def9ite 9tegral1
  !f(x)dx,
is a func;n1 ! antid]ivative ( ;f_4
  ,x's r1sona# & appropriate & 9 fact v
9si<t;l 6ask :yon e>? we use alm ! same
symbolism = two s* radic,y di6]5t no;ns8
,^! two 9tegrals >5't ! same4 ,!y >5't
ev5 ! same k9d ( ?+3 "o is a numb]1 &!
o!r is a func;n4
  ,! ,funda;tal ,!orem ( ,calculus is !
miracul\s !orem t says t ^! two m1n+s (!
9tegral sign >e 9 fact 9timately 3nect$4
,x's wor? ?9k+ l;g 5 ab ! m1n+s (!
symbols t ? 3nec;n 2comes miracul\s1 ev5
?\< t's h>d 2c ! not,n1 9v5t$ af ! !orem
0 prov$1 makes x look n miracul\s1 b
obvi\s4
  ,! 3nec;n cd 2 "w$ \ 9 two ways4 ,f/1
we cd try 6relate ! two types ( 9tegral
0"sh[ mak+ ! def9ite 9tegral---a numb]1
rememb]---96a func;n4 ,!n x wd at l1/ 2
! same type ( object z ! 9def9ite
                                     #ci
9tegral4 ,we d ? 0cr1t+ ! >ea func;n1
  ,f;a"(x) .k !;a^x"f(t)dt_4
,\r f/ =m (! ,funda;tal ,!orem says t
  ,f';a"(x) .k ?d_/dx#!;a^x"f(t)dt
    .k f(x)_4
,9 o!r ^ws1 x says t
  !;a^x"f(t)dt
is an antid]ivative ( f(x)_4
,symbolic,y1
  !f(x)dx .k !;a^x"f(t)dt+c_4
,s once we've turn$ ! def9ite 9tegral
96a func;n1 x turns \ 6be ! same ?+ z !
9def9ite 9tegral4 ,at l1/1 x's ! same ?+
z close z x c 2---x's "o (! possi#
antid]ivatives ( f(x)_4
  ,! second way 63nect !! two types (
9tegral wd 2 6try 6turn ! 9def9ite
9tegral---a func;n---96a numb]1 s t x wd
2 ! same type ( object z ! def9ite
9tegral4 ,! natural way 6turn a func;n
96a numb] is 6take xs value at a po9t4
,= 9/.e1 we turn ! func;n sin  96a numb]
0-put+ sin .p .k #0_4 ,if we want
6produce a numb] t mi<t 2 ! same z !
                                     #dj
numb]
  !;a^b"f(x)dx,
?\<1 we probably ne$ 6"sh[ -b9e ! values
(! func;n at ! #2 po9ts ;a & ;b_4 ,? is
j :at \r second =m (! ,funda;tal ,!orem
says3 ,if
  ,f(x) .k !f(x)dx,
!n
  !;a^b"f(x)dx .k ,f(b)-,f(a)_4
,?us1 turn+ ! 9def9ite 9tegral 96a numb]
0subtract+ xs values at ;b & at ;a makes
x 96exactly ! same ?+ z ! def9ite
9tegral4

,,ftc says 9tegrals & derivatives undo
  "o ano!r4
  ,yet ano!r "u/&+ (! ,funda;tal ,!orem
reg>ds x z cl>ify+ ! s5ses 9 : di6]5ti,n
& 9tegr,n >e 9v]se op],ns ( "o ano!r4 ,9
"o way1 ? is an obvi\s fact1 s9ce !
9def9ite 9tegral is1 af all1 !
antid]ivative4 ,b bo? =ms (! ,funda;tal
,!orem c al 2 9t]pret$ z %[+ s5ses 9 :
di6]5ti,n & def9ite 9tegr,n 9v]t "o
                                     #da
ano!r4
  ,9 \r f/ =m (! ,funda;tal ,!orem1 we
/>t )a func;n f(t)_4 ,we f/ take !
def9ite 9tegral 6get
  !;a^x"f(t)dt_4
,we !n take ! derivative 6get
  ?d_/dx#!;a^x"f(t)dt_4
,! ,funda;tal ,!orem says t ! result is
f(x), : is z close z possi# 6back 6": we
/>t$---! orig9al func;n ) only ! v>ia#
"n *ang$4 ,s ! def9ite 9tegral foll[$ 0!
derivative 9 ? way is ess5ti,y ! id5t;y
op]ator on func;ns4
  ,\r second =m (! ,funda;tal ,!orem is
j a bit m subtle4 ,"h1 we />t )a func;n
,f(x)_4 ,we f/ take ! derivative 6get
f(x) .k ,f'(x)_4 ,we !n take a def9ite
9tegral 6get
  !;a^b",f'(x)dx .k ,f(b)-,f(a)_4
,? isn't q ! same ?+ we />t$ )1 b x's !
close/ we c -e giv5 t we h 65d up )a
numb] & n a func;n4 ,s ? =m (!
,funda;tal ,!orem says t z ne>ly z
possi#1 ! derivative foll[$ 0! def9ite
                                     #db
9tegral is ! id5t;y op]ator on func;ns4

    ,9tegr,n 0,sub/itu;n (,*ange (
    ,v>ia#)
  ,hav+ se5 h[ critical tak+
antid]ivatives is1 bo? 6! pro#m ( -put+
>ea & 6pro#ms 9 physics1 we >e l$
natur,y 6look = me?ods = -put+ !
antid]ivatives ( m func;ns4 ,s f>1 \r
me?ods = -put+ antid]ivatives >e limit$4
,we "k a few basic =mulas1 l
  !x^n"dx .k ?x^n+1"_/n+1#+c
  !sin xdx .k -cos x+c
  !cos xdx .k sin x+c
  !sec^2 xdx .k tan x+c
  !sec xtan xdx .k sec x+c_4
,we al "k "s g5]al facts1 l
  !(f(x)+g(x))dx .k !f(x)dx+!g(x)dx
  !,mf(x)dx .k ,m!f(x)dx_4
,>m$ ) ^! facts1 we c 9tegrate any
polynomial1 & we c h&le "s ?+s 9volv+
trig func;ns4 ,we've al manag$ 69tegrate
a few func;ns 0j play+ >.d4 ,an example
mi<t 2 "s?+ l
                                     #dc
  !cos (2x+3)dx_4
,we wd guess t ! 9tegral mi<t 2 "s?+ l
sin (2x+3)+c_4 ,6te/ ? guess1 we take xs
derivative & see if we get cos (2x+3)_4
,! *a9 rule tells u t
  ?d_/dx#sin (2x+3) .k cos (2x+3)*2_4
,? isn't q :at we 7 look+ =1 b x's
close4 ,if we divide bo? sides by #2, we
get
  ?d_/dx#(?sin (2x+3)_/2#)
    .k ?cos (2x+3)*2_/2# .k cos (2x+3)_4
,x foll[s t
  !cos (2x+3)dx .k ?1_/2#sin (2x+3)+c_4
  ,"! >e pl5ty ( func;ns1 ?\<1 ": ^!
te*niques fall %ort4 ,h[ wd we evaluate
---or c "o evaluate---9tegrals l
  !?xdx_/(x^2"+1)^2"#
    !sin^3 xcos xdx
    !x>x+1]dx
or ev5
  !?dx_/x#=
,x seems cle> t x wd 2 nice 6h m me?ods
( 9tegr,n availa#4

                                     #dd
,! bad news4
  ,hav+ fram$ ? desire1 ,i h 6tell y t
"! is "s bad news 9volv$ ) 9tegr,n4
,9tegr,n is h>d] ?an di6]5ti,n4 ,we're n
go+ 6get rules l ! product & quoti5t &
*a9 rule t let u 9tegrate any func;n
:at"e4 ,"! turn \ 6be simple 9tegrals l
  !sin (x^2")dx
t provably _c 2 express$ 9 t]ms (
ele;t>y func;ns4 ,"! al >e 9tegrals t >e
/>tl+ly -plex1 or />tl+ly unexpect$3
  !cos^5 xdx .k sin x-?2_/3#sin^3 x+?1_/
    5#sin^5 x+c
  !?dx_/1+x^2"# .k arctan x+c_4
,if y're expect+ 9tegr,n 6"w \ z simply
z di6]5ti,n did1 y're expect+ too m*4 ,t
2+ sd1 "! is / a lot we c d 6make
9tegr,n m practical4

,9tegr,n 0,sub/itu;n4
  ,! bi7e/ te*nique = 9tegr,n is call$
,sub/itu;n1 or ,*ange ( ,v>ia#4 ,we "k t
e di6]5ti,n =mula is al an 9tegr,n
=mula2 9tegr,n 0sub/itu;n turns \ 6be !
                                     #de
9tegral =m (! ,*a9 ,rule4
  ,! ,*a9 ,rule says t
  ?d_/dx#(f(u(x))) .k f'(u(x))u'(x)_4
,turn+ ? 96an 9tegr,n =mula gives
  !f'(u(x))u'(x)dx .k f(u(x))+c_4
,? is ! c5tral =mula = 9tegr,n
0sub/itu;n4
  ,n[1 x looks at f/ gl.e z ?\<
sub/itu;n mi<t 2 alm -pletely use.s4 ,af
all1 h[ (t5 wd we expect 65c.t] an
9tegral ": ! 9tegr& j happ5$ 6h ! =m
f'(u(x))u'(x)_8 ,ev5 if an 9tegr& did h
? =m1 cd we reliably recognize t fact8
,x's h>d 6be optimi/ic4
  ,9 fact1 ! situ,n isn't q s bl1k4 ,=
9/.e1 look at ! 9tegral
  !sin^3 xcos xdx
    .k !(sin x)^3"cos xdx_4
,? 9tegral wd fit ! =m ( \r sub/itu;n
=mula if we took
  u(x) .k sin x
  u'(x) .k cos x
  f'(x) .k x^3_4
,69tegrate x1 we'd only ne$ 6f9d
                                     #df
  f(x) .k !f'(x)dx .k !x^3"dx
    .k ?x^4"_/4#+c_4
,!n
  !sin^3 xcos xdx .k f(u(x))
    .k ?u(x)^4"_/4#+c
    .k ?1_/4#sin^4 x+c_4
  ,a second example ( sub/itu;n mi<t 2
"s?+ l ?3
  !#3x^2">x^3"+1]dx_4
,9 ? case1 we cd take
  u(x) .k x^3"+1
  u'(x) .k #3x^2
  f'(x) .k >x]_4
,9 ord] 6get ! 9tegral1 we f/ -pute
  f(x) .k !f'(x)dx .k !>x]dx
    .k ?2_/3#x^3_/2"+c_4
,giv5 ?1 we c say
  !#3x^2">x^3"+1]dx .k f(u(x))+c
    .k ?2_/3#u(x)^3_/2"+c
    .k ?2_/3#(x^3"+1)^3_/2"+c_4
  ,is x cle> t we >e ga9+ "s?+ 0hav+
sub/itu;n ab8


                                     #dg
,ano!r ,view ( ,sub/itu;n4
  ,"! is ano!r way 6look at 9tegr,n
0sub/itu;n t is (t5 easi] 6apply1 ?\< x
takes a mo;t 6see t x makes s5se4 ,we >e
/>t+ \ )! 9tegral (a func;n ( ;x_4
  !f'(u(x))u'(x)dx_4
,if 0magic we cd turn ? 96an 9tegral (a
func;n (a v>ia# ;u,
  !f'(u)du,
!n we'd h an easy answ]3 ! antid]ivative
(a derivative is ! orig9al func;n back
ag1 s
  !f'(u)du .k f(u)+c_4
,if we !n rememb]$ t ;u 0 re,y a func;n
u(x), we'd get ! answ]1
  !f'(u(x))u'(x)dx .k !f'(u)du
    .k f(u)+c .k f(u(x))+c_4
  ,? is j ! =mula = 9tegr,n 0sub/itu;n2
s :at sub/itu;n tell u is t ? magic is
all ju/ifi$4
  ,f ? p]spective1 sub/itu;n is re,y a
me?od = sub/itut+ ;u = u(x), or = *ang+
an 9tegral ) v>ia# ;x 96an 9tegral )!
v>ia# ;u_4 ,? is ": ! "n1 8,*ange (
                                     #dh
,v>ia#0 -es f4
  ,n[1 "h's ": ! 9t]e/+ =malism -es 94
,! orig9al 9tegral1
  !f'(u(x))u'(x)dx,
3ta9s bo? ;x & dx_4 ,9 ord] 6rewrite x z
an 9tegral 9 ! v>ia# ;u, we h 6get rid (
bo? ! ;x &! dx, & 6replace !m ) ;u &
du_4 ,! f/ "p ( ? is simple---we pick !
func;n u(x) t we >e go+ 6use = ;u, & !n
we write ;u 9 place ( ? func;n :]"e x
appe>s4 ,? wd turn ! 9tegral 9to
  !f'(u)u'(x)dx_4
  ,we / h 6get ! dx \ &a du 94 ,we d ?
0! =mal 3/ruc;n
  ?du_/dx# .k u'(x)
  du .k u'(x)dx_4
,? seems 6say x's ,,ok 6replace u'(x)dx
) du_4 ,if we d ?1 lo & 2hold1 we get
  !f'(u)du,
: we c !n rewrite z
  f(u)+c .k f(u(x))+c,
j :at we want$4


                                     #di
,is ? legit8
  ,x's "r 6be suspici\s (! "s:at crazy
look+ algebra abv1 ": we move f
?du_/dx# .k u'(x) to du .k u'(x)dx_4 ,af
all1 ?du_/dx# isn't a frac;n2 x's j a
not,n =! derivative4 ,h[ on e>? c we
ju/ify tak+ x a"p 0multiply+ by dx_8
,:at k9d (a ?+ is dx, anyway8 ,h
ma!maticians flipp$8
  ,? "q is a deep & s]i\s "o t has be5 )
calculus f xs 9cep;n4 ,"! >e a numb] (
ways 6answ] x4 ,on ! simple/ level1 "o
cd say t ? crazy look+ algebra is j a
3v5i5t mnemonic %orth&4 ,we've ju/ifi$
,sub/itu;n 0! use (! ,*a9 ,rule2 s !
valid;y (! me?od doesn't dep5d on !
m1n+;l;s ( ? algebra4 ,we're j us+ !
algebra 2c we h %[n t x gives ! same
result t we wd get 0,sub/itu;n1 b t us+
? %orth& we c get ! result m qkly &
memorably4 ,x's a 3v5i;e1 n a !ory4
  ,? is ! answ] ,i 0 tau<t z a /ud5t1 b
x doesn't seem -pletely satisfactory4
,we'll use calcul,ns l ? s m* t x seems
                                     #ej
depress+ & vaguely 4h"o/ j 6%rug \r
%\ld]s & say t !y >e m1n+.s4 ,isn't "!
any?+ 2h !m8 ,leibniz & two c5turies,8
wor? ( 8 su3essors wd h answ]$ t "!
c]ta9ly is "s?+ 6? algebra4 ,= ,leibniz1
rememb]1 ?du_/dx# re,y 0 a frac;n1 !
ratio ( an 9f9itesimal .,du ov] an
9f9itesimal .,dx_4 ,s9ce ? ?+ t looks l
a frac;n re,y is a frac;n1 multiply+ 0!
denom9ator is c]ta9ly legitimate2 & all
! algebra abv makes s5se4 ,? p]spective
rema9s ! mo/ -mon 3ceptual /&po9t ( mo/
non-ma!matician us]s (! calculus ev5 td4
  ,( c\rse1 "! >e r1l pro#ms ) ,leibniz
,8 answ]4 ,x's n at all cle> :at exactly
^! 9f9itesimals >e4 ,9de$1 if y ?9k
c>e;lly ab :at ! r1l numb]s >e1 x 2comes
m & m cle> t quantities func;n+ l
,leibniz0' 9f9itesimals _c exi/4 ,? 0
alr app>5t at ! "t (! 9v5;n (! calculus4
,bi%op ,b]keley fam\sly describ$
9f9itesimals z 8! <o/s ( de"p$
quantities10 d1d1 t x1 b n q g"o4 ,he
oppos$ calculus 2c ( xs lack ( =mal
                                     #ea
rigor1 >gu+ t calculus l$ 6a!ism4 ,"o
/>t$ \ 02liev+ crazy >gu;ts 9 r1l
analysis1 ,b]keley ?"\1 & :o knew ": x
wd /op8
  ,n[1 ,i don't ?9k calculus repres5ts a
?r1t 6/ud5ts,8 et]nal salv,n1 or ,i wd h
f.d "s o!r l9e ( "w2 b ,i d ?9k
,b]keley's 3c]ns ab ! logical pro#ms (
9f9itesimals >e -pletely legitimate4
,9de$1 calculus 0 a hugely su3ess;l set
( te*niques /&+ on deeply 41s$ f.d,ns
until ! develop;t ( limits 9 ! #19th
,c5tury4 ,! new f.d,ns = calculus 9 t]ms
( limits1 ?\<1 didn't *ange ! results (
!calculus1 only _! 3ceptual "u/&+4 ,!
new logical f.d,ns left 9tact ! old
3clu.ns bas$ on ,leibniz0' 9f9itesimals4
,s ano!r answ] 6! "q (! legitimacy ( \r
crazy algebra is 6say1 calcul,ns l ? >e
,,ok 2c ! c>e;l logical f.d,ns =
calculus 9 t]ms ( limits 7 c>e;lly
design$ precisely 65a# crazy >i?metic l
? 6"w4 ,use x 9 gd h1l?6

                                     #eb
  ,2f ,i l1ve ? topic & we />t look+ at
examples1 let me m5;n two m mod]n
approa*es 6mak+ s5se ( dx & du_4 ,"o
approa* is due 6,abraham ,rob9son 9 !
late #1960_'s_4 ,rob9son 0 a# us+ cl"e
id1s 9 ma!matical logic 63/ruct models
(! r1l numb]s 9 : e !orem y "k ab ! r1ls
/ays true1 b 9 : "! >e actual
9f9itesimals (and _! reciprocals1 : >e
9f9ite numb]s)_4 ,? s-call$ ,non/&>d
,analysis m1nt t =! f/ "t1 "o cd use
9f9itesimals ) -plete rigor & )\t
apology4 ,x 0 a v unexpect$ & b1uti;l
develop;t4
  ,"! is al an approa* 69tegr,n 9 : dx &
du >e reg>d$ n z t9y numb]s1 b z a
"picul> type ( func;ns call$ di6]5tial
=ms4 ,i won't say m ab h[ t "ws "h1 b y
mi<t meet ^! id1s 9 ,multiv>iate
,calculus4

,examples4
  ,n[ t we've giv5 "s sort ( def5se = \r
crazy algebra1 let's try x \ on !
                                     #ec
9tegral
  !#3x^2">x^3"+1]dx_4
,we'd say1 we're go+ 6d ! *ange ( v>ia#
  u .k x^3"+1,
  ?du_/dx# .k #3x^2
  du .k #3x^2"dx_4
,\r goal is n[ 6use ^! equ,ns 6rewrite !
9tegral 9 t]ms ( ;u & du_4 ,we d ?
0replac+ >x^3"+1] by ;u, & replac+
#3x^2"dx by du, 6get ! 9tegral
  !>u]du .k ?2_/3#u^3_/2"+c_4
,we !n rewrite ! result 9 t]ms (! old
v>ia# ;x 6get
  ?2_/3#(x^3"+1)^3_/2"+c_4
  ,! f9e ?+ ab ? approa* is t any sort (
algebra on ! equ,ns = ;u & du is
legitimate1 : (t5 simplifies ! task (
do+ ! sub/itu;n4 ,= 9/.e1 suppose we
want$ 6evaluate ! 9tegral
  !xsin (x^2")dx_4
,we cd try ! sub/itu;n
  u .k x^2
  du .k ?du_/dx#dx .k #2xdx_4
,? m1ns t we c rewrite sin (x^2") z
                                     #ed
sin u, & t xdx .k ?1_/2#du, s t !
9tegral is
  !?1_/2#sin udu .k -?1_/2#cos u+c
    .k -?1_/2#cos (x^2")+c_4

,h[ d y pick ! "r ;u_8
  ,t's ": ! skill -es 94 ,x's :y 9tegr,n
is actu,y fun3 y may h 6try "s di6]5t
possibilities1 & 6be cl"e4 ,"! >e a few
rules ( ?umb "o c use1 ?\<4
  #1_4 ,y "\ 6pick u(x) 6be a func;n t
appe>s 9side ano!r func;n1 s9ce t's !
way x appe>s 9 f(u(x))_4
  #2_4 ,y "\ 6pick u(x) 6be a func;n ^:
derivative u'(x) (or "s?+ close to that)
al appe>s "s": 9 ! 9tegral4 ,af all1 we
want 65d up 9tegrat+ f'(u(x))u'(x)_4
  #3_4 ,"o way 6f9d u(x) is "s"ts 6look
at ! 9tegral & say1 8,:at's ! wor/ ?+ ab
? 9tegral80 ,y !n let u(x) 2 t wor/ ?+1
& pre/o1 x's g"o6
  #4_4 ,f9,y1 y c j try any?+ = u(x) &
see if x "ws4 ,"! >5't _m possibilities
9 a typical 9tegral1 & x doesn't take
                                     #ee
l;g 6try !m all4
  ,^! pr9ciples / don't let y pick ! "r
;u 9 e s+le case4 ,wd x o3ur 6y1 = 9/.e1
t ! "r *oice ( ;u 9 ! 9tegral
  !>1-x^2"]dx
is u .k arcsin x_8 ,b 9 mo/ cases1 "o or
ano!r ( ^! rules ( ?umb w po9t y 9 ! "r
direc;n4

,m examples4
  ,giv5 ! 9tegral
  !?x+3_/(x^2"+6x-5)^2"#dx,
:at wd "o d8 ,a natural *oice wd 2 6let
  u .k x^2"+6x-5
  du .k ?du_/dx#dx .k (2x+6)dx_4
,? *oice cd 2 motivat$ 0pretty m* any (
\r rules ( ?umb4 ,once mak+ x1 we wd say
  ?1_/(x^2"+6x-5)^2"# .k ?1_/u^2"#
  (x+3)dx .k ?1_/2#du_4
,! 9tegral is "!=e
  !?x+3_/(x^2"+6x-5)^2"#dx
    .k ?1_/2#!?du_/u^2"# .k -?1_/2u#+c
    .k -?1_/2(x^2"+6x-5)#+c_4

                                     #ef
  ,"s"ts "o has m ?an "o *oice (a
sub/itu;n4 ,3sid]1 = 9/.e1 \r old fr
  !cos xsin xdx_4
,"o cd d ? 0say+1 let
  u .k cos x
  du .k ?du_/dx#dx .k -sin xdx_4
,we c "!=e rewrite cos x z ;u, & sin xdx
z -du_4 ,! 9tegral is ?us
  !cos xsin xdx .k -!udu
    .k -?u^2"_/2#+c
    .k -?1_/2#cos^2 u+c_4
  ,an equ,y gd *oice wd h be5 6let
  u .k sin x
  du .k ?du_/dx#dx .k cos xdx_4
,we c "!=e rewrite sin x z ;u, & cos xdx
z du_4 ,! 9tegral is ?us
  !cos xsin xdx .k !udu .k ?u^2"_/2#+c
    .k ?1_/2#sin^2 u+c_4
  ,did we get ! same answ]8 ,yes1 s9ce
  ?1_/2#sin^2 x .k ?1_/2#(1-cos^2 x)
    .k -?1_/2#cos^2 x+?1_/2#
    .k -?1_/2#cos^2 x+c_4
  ,= "s 9tegrals1 we ne$ sub/itu;ns t
seem too simple 6be help;l4 ,"h's an
                                     #eg
example ": a v g5tle t\* is all t's
requir$3
  !x>x+1]dx_4
,! wor/ ?+ ab ? 9tegral is ! x+1 "u !
squ>e root4 ,except = t1 we cd j d x4 ,s
let's set
  u .k x+1
  du .k ?du_/dx#dx .k #1 dx .k dx_4
,"p (! 9tegral is n[ easy 6rewrite 9 !
new v>ia#4 ,we replace >x+1] ) >u], & we
replace dx ) du_4 ,b :at d we d )! ;x_8
,well1 we sd 2f t algebra is alw
legitimate2 s if u .k x+1, !n x .k u-1_4
,! 9tegral is !n
  !x>x+1]dx .k !(u-1)>u]du_4
,x takes a mo;t 6r1lize t ? is an
improve;t1 b x is3 we c j multiply \ !
new 9tegral 6get
  !(u>u]->u])du
    .k !(u^3_/2"-u^1_/2")du
    .k ?2_/5#u^5_/2"-?2_/3#u^3_/2"+c
    .k ?2_/5#(x+1)^5_/2"-?2_/3#(x+
      1)^3_/2"+c_4
,? is h>dly ! f/ ?+ "o wd h writt5 d[n 9
                                     #eh
solv+ ! 9tegral4 ,x's wor? di6]5tiat+ !
answ]1 & mak+ sure y re,y d get x>x+1]_4

,us+ sub/itu;n rep1t$ly1 or 6simplify an
  9tegral4
  ,x happ5s "s"ts t "o ne$s 6use 9tegr,n
0sub/itu;n m ?an once 6"w \ an 9tegral4
,= 9/.e1 suppose y 7 3front$ )! 9tegral
  !(2x+5)sin^4 (x^2"+5x-3)cos (x^2"+5x
    -3)dx_4
,a r1sona# sub/itu;n wd 2
  u .k x^2"+5x-3
  du .k ?du_/dx#dx .k (2x+5)dx_4
,af all1 x^2"+5x-3 is obnoxi\s1 & x's
9side ! trig func;ns1 & xs derivative1
#2x+5, is al pres5t 9 ! 9tegral4 ,:5 we
rewrite ! 9tegral1 we get
  !sin^4 ucos udu_4
  ,! new 9tegral is way simpl] ?an ! old
"o1 b x's / n s simple we c d x )\t
ano!r sub/itu;n4 ,s :at ab lett+
  w .k sin u
  dw .k ?dw_/du#du .k cos udu_4
,) ? sub/itu;n1 ! 9tegral 2comes
                                     #ei
  !w^4"dw
    .k ?w^5"_/5#+c
    .k ?sin^5 u_/5#+c
    .k ?sin^5 (x^2"+5x-3)_/5#+c,
": we've "w$ \r way backw>ds "? ! *anges
( v>ia# 6>rive at a func;n ( ;x_4
  ,did we re,y ne$ #2 sub/itu;ns "h8
,no4 ,we cd h us$ a s+le sub/itu;n1
  u .k sin (x^2"+5x-3)
  du .k ?du_/dx#dx .k cos (x^2"+5x-3)*(
    2x+5)dx_4
,? wd h d"o ! 9tegral 9 "o /ep1 _h "o
be5 9si<t;l 5 6see x f ! 2g9n+4
  ,x al "s"ts happ5s t we 5d up d1l+ )
an 9tegral we c't -pletely "w \1 b t a
sub/itu;n c simplify4 ,= 9/.e1 suppose
we ne$$ 6evaluate
  !?sec^2 xdx_/1+tan^2 x#_4
,a r1sona# />t wd 2 6try ! sub/itu;n
  u .k tan x
  du .k ?du_/dx#dx .k sec^2 xdx_4
,! 9tegral !n 2comes
  !?du_/1+u^2"#_4
,at ? po9t1 we're /uck = "r n[4 ,we
                                     #fj
don't "k h[ 6d ! 9tegral we're left )4
,b if we h 6approximate ! answ]
num]ic,y1 wdn't y r "w )! 9tegral 9 ? =m
?an 9 ! o!r =m8
  ,9cid5t,y1 "! is a bett] way 6d !
9tegral4 ,rememb] t
  sin^2 x+cos^2 x .k #1
  ?sin^2 x_/cos^2 x#+?cos^2 x_/cos^2 x#
    .k ?1_/cos^2 x#
  tan^2 x+1 .k sec^2 x_4
,! 9tegral is ?us
  !?sec^2 xdx_/1+tan^2 x#
    .k !?sec^2 xdx_/sec^2 x# .k !dx
    .k x+c_4
,? actu,y l1ds u 6an 9t]e/+ & surpris+
obs]v,n4 ,we've j se5 t if u .k tan x,
!n
  !?du_/1+u^2"# .k x+c .k arctan u+c,
: is equival5t 6! derivative =mula
  ?d_/du#arctan u .k ?1_/1+u^2"#_4
  ,c y prove ? =mula )\t us+ 9tegr,n
0sub/itu;n8


                                     #fa
,9tegrals ne$+ "s algebra f/4
  ,let's look ag at ! 9tegral
  !cos^5 xdx,
= : ,i pull$ a value \ 0magic at ! 2g9n+
( ? sec;n4 ,i cd ,i h d"o ?8 ,"! doesn't
seem 6be an obvi\s sub/itu;n = ?
9tegral1 b "h's a lovely trick3 ,y "k t
cos^2 x .k #1-sin^2 x, : implies t
cos^4 x .k (1-sin^2 x)^2_4 ,! 9tegral is
?us
  !(1-sin^2 x)^2"cos xdx_4
  ,6d ?1 wdn't "o natur,y take
  u .k sin x
  du .k ?du_/dx#dx .k cos xdx_4
,! 9tegral wd !n 2come
  !(1-u^2")^2"du
    .k !(1-2u^2"+u^4")du
    .k u-?2_/3#u^3"+?1_/5#u^5"+c
    .k sin x-?2_/3#sin^3 x+?1_/5#sin^5 x
      +c,
! same my/]i\s answ] m5;n$ ne> ! 2g9n+ (
^! notes4
  ,? same te*nique "ws = e odd p[] (
sin x or ( cos x_4
                                     #fb
,sub/itu;n ) def9ite 9tegrals4
  ,s f>1 all \r 9tegrals h be5 9def9ite
9tegrals4 ,:at if we 7 6try do+
sub/itu;n on a def9ite 9tegral8 ,y'd 2
tempt$ j 6d "s?+ wr;g l ?3 ,! sub/itu;n
  u .k x^2"+5x+2
  du .k ?du_/dx#dx .k (2x+5)dx
lets u write
  !;0^1"(x^2"+5x+2)^3"(2x+5)dx
    .k !;0^1"u^3"du
    .k @(?u^4"_/4#@);0^1
    .k ?1^4"_/4#-?0^4"_/4# .k ?1_/4#_4
  ,? c't possibly 2 "r1 ?\<1 s9ce !
m9imum value (! func;n
(x^2"+5x+2)^3"(2x+5) on ! 9t]val
#0 "k: x "k: #1 is #40, s t ! 9tegral
ov] an 9t]val ( l5g? #1 has 6be at l1/
#40_4
  ,:at w5t wr;g8
  ,"o way 6see :y ? calcul,n fail$ is
6"w a bit m c>e;lly4 ,if we 7 do+ !
9def9ite 9tegral1 we cd 9de$ write
  !(x^2"+5x+2)^3"(2x+5)dx
    .k !u^3"du
                                     #fc
    .k ?u^4"_/4#+c
    .k ?(x^2"+5x+2)^4"_/4#+c_4
,! ,funda;tal ,!orem ( ,calculus wd n[
say t s9ce ? is an antid]ivative (!
9tegr&1 we h
  !;0^1"(x^2"+5x+2)^3"(2x+5)dx
    .k @(?(x^2"+5x+2)^4"_/4#@);0^1
    .k ?(1^2"+5(1)+2)^4"_/4#-?(0^2"+5(0)
      +2)^4"_/4#
    .k ?8^4"_/4#-?2^4"_/4# .k #1020_4
  ,we c n[ see ! pro#m ) \r e>li]
calcul,n4 ,we plu7$ 9 u .k #0 & u .k #1,
:5 we %d h plu7$ 9 x .k #0 & x .k #1_4
  ,"o way 6d def9ite 9tegrals )
sub/itu;n is "!=e 6f9d ! 9def9ite
9tegral1 write x back 9 t]ms (! orig9al
v>ia# ;x, & !n plug 9 ! 5dpo9t values =
;x & take ! di6];e4
  ,"h's an equival5t b simpl] proc$ure4
,imag9e we h ! 9itial 9tegral writt5 \ m
explicitly ?an usual1 l ?3
  !;x ;.k #0^x ^.k #1"(x^2"+5x+2)^3"(2x
    +5)dx_4
,! sub/itu;n
                                     #fd
  u .k x^2"+5x+2
  du .k ?du_/dx#dx .k (2x+5)dx
wd let u rewrite ? z
  !;x ;.k #0^x ^.k #1"(x^2"+5x+2)^3"(2x
    +5)dx
    .k !;x ;.k #0^x ^.k #1"u^3"du_4
,ah1 "h's ! pro#m ) \r f/ calcul,n3 ,we
"\ 6*ange ! 5dpo9ts ( 9tegr,n 6reflect !
new v>ia#4 ,h[8 ,well1 :5 x .k #0, we h
u .k #0^2"+5(0)+2 .k #2, & :5 x .k #1,
we h u .k #1^2"+5(1)+2 .k #8_4 ,we %d
"!=e rewrite ! 9tegral z
  !;x ;.k #0^x ^.k #1"u^3"du
    .k !;u ;.k #2^u ^.k #8"u^3"du
    .k !;2^8"u^3"du
    .k @(?u^4"_/4#@);2^8
    .k ?8^4"_/4#-?2^4"_/4# .k #1020_4
  ,/>e at ? a bit until y see :y x is t
! me?od ( runn+ ! 9def9ite 9tegral back
to ;x & !n plu7+ 9 ! old 5dpo9ts = ;x
results 9 exactly ! same result z l1v+ !
9tegral 9 t]ms ( ;u & plu7+ 9 new
5dpo9ts = ;u_4

                                     #fe
  ,"h's ano!r example ( ? sort (
calcul,n1 writt5 ! way "o wd norm,y see
x set d[n4 ,6-pute
  !;0^.p_/2"(cos x)>sin x]dx,
we "\ 6use ! sub/itu;n
  u .k sin x
  du .k ?du_/dx#dx .k cos xdx_4
,! 9tegral is ?us
  !;0^.p_/2"(cos x)>sin x]dx
    .k !;0^1">u]du
    .k @(?2_/3#u^3_/2"@);0^1
    .k ?2_/3#(1^3_/2")-?2_/3#(0^3_/2")
    .k ?2_/3#_4
,! 5dpo9ts *ang$ :5 we did ! *ange (
v>ia# 2c :5 x .k #0, we h
u .k sin x .k sin #0 .k #0, & :5
x .k .p_/2, we h
u .k sin x .k sin ?.p_/2# .k #1_4

,d we ne$ ?8
  ,/ud5ts (t5 don't l ? *ange ( v>ia# 9
def9ite 9tegrals4 ,x's 3fus+ at f/1 & x
seems 6_m p t x's easi] j 6"w \ !
9def9ite 9tegral 9 t]ms ( x, & !n 6plug
                                     #ff
9 values4 ,let me 5d ? sec;n ) an
example %[+ t y c't alw d ?4 ,"s"ts y
re,y ne$ 6*ange ! limits ( 9tegr,n 9 !
def9ite 9tegral4
  ,"h's ! example3 look at
  !;-1^1"2xsin ((x^2"-1)^2")dx_4
,! natural ?+ 6d 6? 9tegral wd 2 6let
  u .k x^2"-1
  du .k ?du_/dx#dx .k #2xdx_4
,! 9def9ite 9tegral wd !n 2come
  !#2xsin ((x^2"-1)^2")dx
    .k !sin (u^2")du_4
  ,! tr\# n[ is t we c't d ! new
9tegral4 ,9 fact1 "! is provably no
9def9ite 9tegral = sin (u^2") t c 2
writt5 9 t]ms ( ele;t>y func;ns (
polynomials1 trig func;ns1 logs &
expon5tials)_4 ,y're -pletely /uck4
  ,is ! def9ite 9tegral impossi# )\t
resort+ 6num]ical me?ods1 !n8 ,n at all4
,:5 x .k -#1, u .k (-1)^2"-1 .k #0_2 &
:5 x .k #1, u .k #1^2"-1 .k #0_4 ,!
def9ite 9tegral is "!=e
  !;-1^1"2xsin ((x^2"-1)^2")dx
                                     #fg
    .k !;0^0"sin (u^2")du .k #0,
s9ce ! >ea (a l9e seg;t is #0_4 ,3clu.n3
y d ne$ 6-e 6t]ms ) *ange ( v>ia# 9
def9ite 9tegrals4 ,"s"ts x's ! only game
9 t[n4



















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