Indefinite Integration - Classwork J;K4!;!A*494!&8!+&)43&&K!A;A45!;+0!9&>45!(A!)64!A;5;=5;A6!(+045!)64!96;5)!34'&1.!$4'&1I!)6454!;54!@!)452,.!E5*)4 16;)!D&(!844'!;54!)64!*+>45,4,!&8!4;96!&8!)64,4!)452,. )452 *),!*+>45,4

@ +&+,4+,4

-[B +&+,4+,4

3&D +&+,4+,4

0&= +&+,4+,4

6&)!0&= +&+,4+,4

#,!5*0*9('&(,!;,!)64!';,)!A5&3'42!*,!Q6&1!9;+!D&(!6;>4!;+!*+>45,4!&8!;!6&)!0&=GR!,&!;54!)64D!;''!5*0*9('&(,.!J64 A5&3'42!*,!)6;)!;''!&8!)64,4!)452,!;3&>4,!;54!+&(+,.!C+>45,4,!54845!+&)!T(,)!)&!;+!&AA&,*)4!3()!)&!;+!&AA&,*)4 A5&94,,.!E4!);K4!*+>45,4,!&8!>453,I!+&)!+&(+,.!E5*)4!)64!*+>45,4,!&8!)64,4!A5&94,,4,. A5&94,, *),!*+>45,4

,*)!0&1+ ,);+0!(A

=4)!054,,40 =4)!(+054,,40

);K4!;!3&&K!6&24 );K4!*)!)&!,96&&'

=4)!14) 05D!&88

=&!)&!,'44A 1;K4!(A

C+!2;)642;)*9,I!D&(!6;>4!'4;5+40!;3&()!&A45;)*&+,!&+!8(+9)*&+,! f .!J64!*+>45,4!&A45;)*&+!*,!04+&)40!;,! f !1!Q+&) )&!34!9&+8(,40!1*)6! x !1 I!)64!549*A5&9;'!&8!x!Q54242345!x!*,!;!+&(+!;+0!f!*,!;!A5&94,,!16*96!*,!;!>453.R!!E64+!4>45 D&(!A458&52!;+!&A45;)*&+!;+0!*2240*;)4'D!A458&52!*),!*+>45,4I!D&(!1*''!4+0!(A!4F;9)'D!16454!D&(!,);5)40!1*)6.!E4 ,;D!)6;)! f !1 f # x$ ! x ;+0!;',&! f f !1 # x$ ! x .!!$4'&1!15*)4!,&24!2;)642;)*9;'!8(+9)*&+,!;+0!)64*5!*+>45,4,.

&

&

'

'

/(+9)*&+

C+>45,4

/(+9)*&+

;00*)*&+

,(3)5;9)*&+

'&=

C+>45,4 -7x

2(')*A'*9;)*&+

0*>*,*&+

)5*=

*+>45,4!)5*=

,:(;54

,:(;54!5&&)

5;*,*+=!)&!n!A&145

);K*+=!n)6!5&&)

%&I!&3>*&(,'D!,*+94!0*88454+)*;)*&+!&8!8(+9)*&+,!*,!;!A5&94,,I!14!2(,)!6;>4!;+!*+>45,4!&8!)6;)!A5&94,,.!E4!9;''!)6;) A5&94,,!;+)*0*88454+)*;)*&+.!!!/&5!*+,);+94I!14!K+&1!)6;)!)64!045*>;)*>4!&8! y ! x 3 *,! 3x 2 .!%&!*)!!2;K4,!,4+,4!)&!,;D )6;)!;+!;+)*045*>;)*>4!&8! 3x 2 !*,! x 3 . Z&14>45I!*)!*,!*2A&5);+)!)&!,;D!an!;+)*045*>;)*>4!&8! 3x 2 !5;)645!)6;+!the!;+)*045*>;)*>4!&8! 3x 2 !8&5!)64!,*2A'4!54;,&+ )6;)!)64!045*>;)*>4!&8! y ! x 3 *,! 3x 2 I!3()!,&!*,!)64!045*>;)*>4!&8! y ! x 3 " 2 I! y ! x 3 ! 5I!;+0! y ! x 3 " 6 " .!J6454!;54 ;+!*+8*+*)4!+(2345!&8!8(+9)*&+,!16&,4!045*>;)*>4!*,! 3x 2 .!%&!164+!14!=&!3;9K1;50,!)&!)64!;+)*045*>;)*>4!*)!*, *2A&,,*3'4!)&!04)452*+4!16*96!8(+9)*&+!*)!9;24!85&2.!%&!)&!9&>45!&(5!34),I!14!,;D!)6;)!)64!;+)*045*>;)*>4!&8! 3x 2 !*, x 3 " C I!16454!C!54A54,4+),!;!9&+,);+).!E4!9;''!C!)64!9&+,);+)!&8!*+)4=5;)*&+.!C)!*,!*2A&5);+)!)&!;));96!)64!cC!;8)45 4>45D!;+)*045*>;)*>4.!E6;)!D&(!;54!0&*+=!*,!,;D*+=!)6;)!)64!;+)*045*>;)*>4!*,!;!8;2*'D!&8!8(+9)*&+,!5;)645!)6;+!&+4 ,A49*8*9!8(+9)*&+. J64!A5&94,,!&8!);K*+=!;+)*045*>;)*>4,!*,!9;''40!integrationI!,A49*8*9;''D!indefinite integration!349;(,4!&8!)64!9&+,);+)!&8 *+)4=5;)*&+!C.!%&!14!0&!+&)!6;>4!)&!15*)4!)64!1&50!;+)*045*>;)*>4!;=;*+I!14!(,4!;!,D23&'!)&!54A54,4+)!;+ ;+)*045*>;)*>4.!!J6;)!,D23&'!*,!9;''40!;+!*+)4=5;'!,*=+!16*96!*,!15*))4+!'*K4!)6*,d!!.!!!J64!1;D!14!15*)4!;+!*+)4=5;'!*,d

+ f # x$ dx ! F # x$ " C .!!J64!dx!)4'',!D&(!16;)!)64!*2A&5);+)!>;5*;3'4!*,!164+!D&(!;54!*+)4=5;)*+= $ dy dy ' T(,)!;,!D&(!+440!)&!K+&1!16;)!)64!*2A&5);+)!>;5*;3'4!*,!164+!D&(!0*88454+)*;)4! & as opposed to ) . dx ( % dt www.MasterMathMentor.com !!"!#$!%&'()*&+,!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!"!-4!045*>;)*>4!5('4,I!14!6;>4!;!9&554,A&+0*+=!5('4!8&5!*+)4=5;',.!!Z454!;54!,&24!3;,*9!*+)4=5;)*&+!5('4,. Differentiation formula Integration formula d 0 dx ! C &C ' ! 0 dx d k dx ! kx " C &kx' ! k ! dx d kf # x$ dx ! k f # x$ dx " C !!!"!8;9)&5!&()!9&+,);+) kf # x$ ! kf ,# x$ ;!9&+,);+)!9;+!34!f8;9)&540!&()g dx d f # x$ ( g # x$ dx ! f # x$ dx " g # x$ dx " C f # x$ ( g # x$ ! f ,# x$ ( g ,# x$ dx !!!!!!!)64!045*>;)*>4!&8!;!,(2!*,!)64!,(2!&8!045*>;)*>4, !!!*+)4=5;'!&8!;!,(2!*,!)64!,(2!&8!)64!*+)4=5;', x n"1 d n x n dx ! " C !"!)64!A&145!5('4!54>45,40 x ! nx n!1!!"!)64!A&145!5('4 n "1 dx ]F;2A'4,!"!8*+0!)64!*+)4=5;'!&8!4;96!&8!)64!8&''&1*+=d x6 x13 -R! 7 dx ! 7 x " C 45.!J6454!;54!&+'D!;!945);*+!,4)!&8!5('4,!;+0!*8!;+!*+)4=5;)*&+!A5&3'42 0&4,+\)!8*)!&+4!&8!)64!5('4,I!D&(!2;D!6;>4!)&!96;+=4!)64!4FA54,,*&+!,&!)6;)!*)!0&4,.!C!9;''!)6*,!;!f3;=!&8!)5*9K,.g !!!!!!!!J5*9K!-!"!2(')*A'D!&()I!)64+!*+)4=5;)4!!!!!!!!J5*9K!
! 6x2 " 9x " C

+

x 2 " 3x " 1 dx x4

+ #x

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !1 x

!2

!

" 3 x !3 " x !4 $ dx

3 1 ! 3 "C 2 2x x

-SR

!!

+#

2 x ! 5$#3x " 2$

+ #6 x

x 3

2

1

dx

! 11x 2 ! 10 x

!1

2

$ dx

1 12 5 2 22 3 2 x ! x ! 20 x 2 " C 5 3

www.MasterMathMentor.com !!"!#$!%&'()*&+,!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!"!-4+!; dx dy ,);)424+)!;3&()!)64!045*>;)*>4!&8!yI! .!`&(5!=&;'!*,!)&!,&'>4!8&5!y.!!E4!6;>4!0&+4!,&!1*)6!)64!4F94A)*&+!&8!)64 dx cCI!)64!9&+,);+)!&8!*+)4=5;)*&+.!%&!14!6;>4!;!general solution !&8!)64!N]h.!$()!,(AA&,4!14!1454!)&'0!)6;)!*8!x!b!7I )64+!y!b!@.!!/5&2!)6454!14!9;+!,&'>4!8&5!C!;+0!14!1*''!)6(,!6;>4!)64!specific solution!&8!)64!N]h.!!W4)\,!0&!,&. J64!&5*=*+;'!,);)424+)!

y ! 0 " C ! 5!,&! C ! 5!!!,&! y ! x 2 " 5 ]F;2A'4!-SR!%&'>4!)64!0*88454+)*;'!4:(;)*&+.

]F;2A'4!-UR!%&'>4!)64!0*88454+)*;'!4:(;)*&+.

f ,# x$ ! x 2 ! 2 x " 2, f #3$ ! !1

f ,# x$ ! 3x ! 1, f #2$ ! 3 f # x$ !

f # x$ !

+ #3x ! 1$ dx

+ # x 2 ! 2 x " 2$ dx

x3 ! x2 " 2x " C 3 f # 3$ ! 9 ! 9 " 6 " C ! !1 # C ! !7

3x 2 f # x$ ! !x"C !!!!!!!!!!!!! 2 f #2$ ! 6 ! 2 " C ! 3 # C ! !1

f # x$ !

f # x $ ! 3x 2 ! x ! 1

f # x$ !

]F;2A'4!