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Reference Guide

Calculus Integrals II - Reference Guides

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l e a r n • r e f e r e n c e • r e v i e w permacharts TM CCalculus Integrals II INTEGRALS INVOLVING a+bu & p+qu INTEGRALS INVOLVING a+bu & p+qu du 1  +  pu+ 2( )u+ −p aq ∫( )puq ( ) = − lnab+  +C ∫ du = 2 a++C a+ 3b udu = 1 a l( )+ − +ln( ) C  + du 1  q( )u − − bp  ∫( )+uq ( ) − b q  ∫ = ln  +C ( )+ab qa+q −  q( )u +aq bp  du 1 1  q  +    ∫ 2 =  + ln + C ( ) puq ( ) aq− +  bu aq−bp a+  ab 2 aub 2 bp aq −1 qa( ) ∫ du = − tan +C   puq q qq bp aq udu = 1  p lna+  − a +C ∫( ) pq2( ) aq − −a bp p+  b( )u  n +1 n ∫(pq ) ab = 2( ) aub + aq bp ∫( ) du 2 2 q( )+ 23qn( )+ ab udu = a + ∫( ) puq2( ) bqa( )b ( ) ∫ du = aub + 1 p2 aa( )−2p  ( )q n aub ( ) aq () ( )p qu n 1 2 q ln( ) + 2 ln( ) C  ( ) bp  b  b()3− du 2( ) aq n()− ∫ n 1 du 1  1 ( ) aub ∫ m n =− m  n −1 1 + ( ) puq ( ) ( )−bp ()−1  ) puq ( ) n n n− 1  ∫( )+ du = 2( ) ab +2n( )a− ∫( )+ du du  aub b()1+ bn() + ab b( )+− 2 ∫ m n−1+C ( ) pq ( )  ∫ ab du=− aub + b ∫ du ab du = + aq bp lnp+ C + ( ) n q() ( ) qu n −1 2q()− ( ) n −1 ab ∫ pq q p2 ( ) INTEGRALS INVOLVING a –u , u a 2 2 2 2 INTEGRALS INVOLVING a+bu & p+qu du 1 ua  1 −1u du = +2 ln b( )u +qa ( ) C  + ∫ua2−2 = 2a lnua+  +C a coth a C ∫ ( )pq ( ) bq   udu 1 udu a ua −  ∫ ∫ = − ln() C2 + =u ln + C udu ( )uq ( ) aq bp du ua2−2 2 2 ua−2 ua+  ∫ ∫ = − ( )puq( )+ bq 2bq ( )pq ( ) udu u a2 2 2 ∫ 2 2= +22 − () C + 2bqu aq bp ua − ∫ ( )pud ( ) = ( )puq ( ) − du 1 1  −  4bq ∫ = +2 3 ln  +C aq bp 2 u ()a−2 au 2 u+  ( ) du 8bq ∫ ( )puq( )  2  du = − 1 1 ln u +C ∫uu()− 2 2au22 24 ua−2  ( ) ( )+uq ( ) bp −q du ∫ ∫ du = + ( ) b b ( )puq( ) du =− u − 1 l ua +C ∫ 2 2 2 2au a − 4a3  ua  du 2 ab () − () ∫ = +C ( ) aubp( ) ( ) ( ) aq p qu+ udu 1 ∫ 2 2 2 =− 2 2 +C ()− 2()a− udu u 1  ua ∫ 2 =− + l  +C ua−2 2()a−2 4a  ua  INTEGRALS INVOLVING e u & lnu () 3 2 au 2 3 udu a 1 2 2 e du = l+n au + () ()+ + ∫ 2 2 2 =− 2 ua−2 + 2 l()ua C + ∫ u 11 !2 2⋅ 3 3⋅ () − () au a au e du =− e + a e du du 1 1  u2  ∫ un ()−1 n−1 n −1 un−1 ∫ =− + ln + C uu 2 − 2 2au()2−2 2a4 ua−2  () ∫ du = − 1+l( )q Cau pe+ au pqa du =− 1 − u − 3 ln −  +C ∫ 22 2 2 4 au 2au a − 4a5 ua+  du u 1 1 au uu() − www.p ermach()ts .co m ∫ 2 = +2 2 a − + l( ) qe C + ( )+ au p ap( )qe ap du 1 1 1  u2  ∫ 22 4 − + 6 nn 2 2 +C a n −1 2 uu a−2 2au 2au()−2 a  ua−  au n u e bsin n()− b au n−2 () u d ue∫b s∫n = abn+ 22 ( )ub − + cos uab+d 2 2 e bsin du =− u − 2n − du au n−1 n()− b 2 ∫ 2 2 n 2 n 2 2 n −1 2an( )− 2 2 1 u d u e bosn = e bcos ( )s + +sin u 2 22 u∫e bcosn−2 ()a− 2an()u a() − ()a− ∫ abn+2 ab+ udu 1 au e u ln 1 ea1 lnu lnu ∫ 2 2 n =−n 2 2 −1+C u∫ ∫∫ ln = − du+ − = 2du C () − n u2()− () − a a u u u u n n +1 du 1 1 du ln udu = ln u−+ ≠, 1 = + du ln lnC ∫ n =− n−1 2 ∫ 1 ∫ u∫ n+1 uln () u()2−a 2 1()u a() a u()2 − lnnudu = lnu n ln −udu m m−2 m−2 ∫ ∫ udu = ud +a2 ud ∫ ∫2 2 ∫n 2 2 n−1 2 2 n mn u unln n m n −−1 () − ()a− () − ∫undu = m +1 1 −m + ∫u ln udu, m ≠− 1 du = 1 1 du − du n 2 2 3 3 ∫ m 2 2 n a 2 m −22 2 n a2 ∫ m 2 2 n 1 udu ()n+u1u l( ) +1 ln uu () − u ua () − uu() − ∫ lnu =+n()( )n u 1 ln 2⋅ ! + 33⋅ ! + GENERAL NOTES • u is a function of x • Inverse trigonometric and hyperbolic functions represent principal • a, b, m, n, p, and q are real constants (restricted where indicated or values necessary) • All denominators are assumed to not equal zero • C represents the constant of integration • This reference guide may be used by itself or in conjunction with Calculus Integrals I • All angles are measured in radians 2 CALCULUS INTEGRALS II • 1-55080-799-4 w w w.permacharts.com © 1996-2013 Mindsource Technologies Inc. l e a r n • r e f e r e n c e • r e v i e w permachartsM TRIGONOMETRIC INTEGRALS CONT’D du =− 1 cotau C du =− cosau + −1 ln tanu + C = sinaudu sinau + a cosaudu 2sin2au 3a ∫∫ sin au 2auain 2a2   un n n () u1 −1 −n− 1 u udu u π au 2   au  du 1 π πau 1 3  au π du 1  2 ∫1 −sinau = atan + a2lnsi42 +C ∫ ∫ 2 = 2a2tan + 6a4tan  1 +C −sinau = atan42   ( )sinau du cosau n− 2 du udu ucaos 1 n −2 udu ∫ ∫ ∫ n =− n−1 n + n −1 −2 =− n−1 − 2 2 2 n− n− +n −1 ∫ sin au an()−1 sin au sin au sinau a()− 1sin au a n () ()2 − sinau sin au du 1 du 1 au ∫ ∫ 2 = + tanau C =− cot + C du = + sina+ 1 ln tanπ au +C cos au a 1 −cosau a 2 ∫cos3au2
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