Integral Table Master

Integral Table

TABLE OF ITEGRALS

Remark:

letter a, b, c, α and β denote contants; n denote positive integer, u denote variabl

Emath2, Emath3, Am1, Am2

0.

⌠   ⌡

ln( u) du → u⋅ ln( u) − u

⌠ ( n+ 1) u  n 1.  u du → ( n + 1) ⌡ ⌠  2.   ⌡ ⌠  3.  ⌡

4.

⌠   ⌡

1 u

right arrow means result from mathcad = means result from direct input

n ≠ −1

du → ln( u)

exp( u) du → exp( u)

u α du →

1 ln( α )

⋅α

u

⌠  5.  ⌡

sin( u) du → −cos ( u)

⌠  6.  ⌡

cos ( u) du → sin( u)

⌠  2 10.  csc( u) du = − cot( u) ⌡

⌠  11.  ⌡

sec ( u) ⋅ tan( u) du → sec ( u)

⌠  12.  ⌡

csc ( u) cot( u) du → −csc ( u)

⌠  13.  ⌡

sec ( u) du → ln( sec ( u) + tan( u) )

⌠  14.  ⌡

csc ( u) du → ln( csc ( u) − cot( u) )

⌠  15.    ⌡ ⌠  16.    ⌡

⌠  7.  tan( u) du → − ln( cos ( u) ) ⌡ ⌠  8.  cot( u) du → ln( sin( u) ) ⌡

17.

1

 u  α

du = asin

2 2 α −u

1 2 α +u

du →

2

⌠   A dB = AB − ⌡

1 α

 u  α

⋅ atan 

⌠   B dA ⌡

⌠  2 9.  sec ( u) du = tan( u) ⌡

Integral Table Master.mcd

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Integral Table

Forms Involving α + β ⋅ u

18.

19.

20.

⌠ 1  n n+ 1  ( α + β ⋅ u) du = β ( n + 1) ⋅ ( α + β ⋅ u) ⌡ ⌠    ⌡ ⌠    ⌡

⌠  21.    ⌡

1 α + β ⋅u

du →

⌠  26.   ⌡

n ≠ −1

1

β

du =

1 α

 u    α + β ⋅u

⋅ ln

28. 1 2 u ⋅ ( α + β ⋅ u)

du = −

1

+

α⋅u

β α

2

 α + β ⋅u   u 

23.

24.

25.

⌠    ⌡ ⌠     ⌡

1 u⋅ ( α + β ⋅ u)

2

du =

1 α

⋅ ( α + β ⋅ u) −

 α + β ⋅u ⋅ ln  2  u  a 1

30. u α + β ⋅u

du →

u ( α + β ⋅ u)

⌠   u⋅ ⌡

2

1 β

du →

α + β ⋅ u du →

Integral Table Master.mcd

⋅u −

α β

2

⋅ ln( α + β ⋅ u)

α 2

⌠  2  u ⋅ ⌡ ⌠     ⌡

7  2 2 ⋅  ⋅ ( α + β ⋅ u) 3 7 β 1

α + β ⋅ u du →

2 u

du →

α + β ⋅u

− 2⋅ ( α

5  2 2 ⋅  ⋅ ( α + β ⋅ u) 3 5 β 1

−

4

−

3

4 5

⋅α

⋅ ln

29. ⌠  22.    ⌡

du →

α + β ⋅u

3  2 2 ⋅  ⋅ ( α + β ⋅ u) 2 3 β 1

ln( α + β ⋅ u) 27.

u⋅ ( α + β ⋅ u)

u

β ⋅ ( α + β ⋅ u)

+

1 β

2

5  1 2 2 ⋅  ⋅ ( α + β ⋅ u) 2 5 β

3 2 3

⋅ α ⋅ ( α + β ⋅ u)

⌠    ⌡

⌠  31.   ⌡

⋅ ln( α + β ⋅ u)

−

⌠    ⌡

2

  

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⌠  32.    ⌡

1 u⋅

α + β ⋅u

1 u⋅

du =

  

⋅ ln

1 α

2

du →

1

α + β ⋅u ( −α )

α + β ⋅u u

du = 2⋅

1 2 u ⋅

α + β ⋅u

du =

2

1   ( α + β ⋅ u) 2 ⋅ atan  1   ( −α) 2 

α + β ⋅u α ⋅u

α 

α + β ⋅u +

⌠  α + β ⋅ u + α ⋅  ⌡ −

α 

α + β ⋅u −

u α + β⋅

⌠  − ⋅ 2⋅ α   ⌡ β

u⋅

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Integral Table

2 2 2 2 Forms Involving u + α , u − α

33.

⌠     ⌡

⌠  34ii.    ⌡ ⌠  34ii.    ⌡

35.

⌠     ⌡

1 2 2 u −α

u 2 2 u +α

u 2 2 u −α

2 u 2 2 u −α

du =

du →

du =

1 2⋅ α

1 2

1 2

⌠  37i.    ⌡

 2 2 ⋅ lnu + α 

⌠  37ii.    ⌡

 2 2 ⋅ lnu − α 

du = u +

Integral Table Master.mcd

⌠  36.    ⌡

u − α  u + α

⋅ ln

α 2

2 u 2 2 u +α

 u  α

du = u − α ⋅ atan 

1

 2 2 u⋅ u + α 

1

 2 2 u⋅ u − α 



1

du =

2⋅ α

du = −

2

   2 2 u + α  

1 2⋅ α

2 u

⋅ ln

2

   2 2 u − α 

⋅ ln

2 u

u − α  u + α

⋅ ln

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Integral Table

2 2 u +α ,

Forms Involving

2 2 u −α

⌠  38i.   ⌡

u 2 2 u + α du = ⋅ 2

2  2 2 α u +α + ⋅ lnu + 2

2 2 u +α 

⌠  38ii.   ⌡

u 2 2 u − α du = ⋅ 2

2  2 2 α u −α − ⋅ lnu + 2

2 2 u −α 

⌠   43ii.    ⌡ ⌠  44.    ⌡

3

⌠  39i.  u⋅  ⌡

1  2 2 2 2 u + α du = ⋅ u + α  3

2

⌠  45.    ⌡

3

⌠  39ii.  u⋅  ⌡

1  2 2 2 2 u − α du = ⋅  u − α  3

2

⌠  2 40i.  u ⋅  ⌡

u  2 2 2 2 u + α du = ⋅ 2u + α  ⋅ 8

4  2 2 α u +α − ⋅ ln u + 8

2 2 u +α

 

⌠  2 40ii.  u ⋅  ⌡

u  2 2 2 2 u − α du = ⋅ 2u − α  ⋅ 8

4  2 2 α u −α − ⋅ ln u + 8

2 2 u −α

 

⌠   41.   ⌡ ⌠   42.   ⌡ ⌠   43i.    ⌡

2 2 u −α u

2 2 u +α u

2 2 u +α 2 u

2 α 2 u − α − α ⋅ acos  u

du =

du =

Integral Table Master.mcd



2 2 u +α 



u

α + 2 2 u + α − α ⋅ ln

du =

−

2 2 u +α u

  



+ lnu +

⌠  46i.    ⌡ ⌠  46ii.    ⌡ ⌠   47i.    ⌡

 

⌠   47ii.    ⌡

2 2 u +α 

Copyright 2009

2 2 u −α 2 u

1

du =

2 2 u −α

−

u



+ lnu +



2 2 u +α 



2 2 u −α 

du = lnu +

2 u −

2 2 u +α

1

du = lnu +

2 2 u −α

u

du =

2 2 u +α

du =

2 2 u −α

2 2 u +α

u 2 2 u −α

2 u

du =

2 2 u +α

2 u 2 2 u −α

du =

u 2

u 2

⋅

2  2 2 α u +α − ⋅ ln u + 2

⋅

2  2 2 α u −α + ⋅ ln u + 2

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Integral Table

⌠  48.    ⌡ ⌠  49.    ⌡ ⌠  50i.    ⌡ ⌠  50ii.    ⌡ ⌠  51.    ⌡ ⌠  52.    ⌡ ⌠    53i.  ⌡ ⌠    53ii.  ⌡

1 u⋅

du =

2 2 u −α

1 u⋅

du =

2 2 u +α

1 2 u ⋅

3 u ⋅



α

u

 α +

−

  2 2 u +α 

⌠  54ii.      ⌡

2 2 u +α

⌠  55i.      ⌡

2 2 u −α 2 α ⋅u

2 2 u −α

du =

2 2⋅ α ⋅ u

2 2 u −α

−

+

2 2 u +α 2 2α ⋅ u

2 2 u +α

⋅ acos 3  2α 1

+

α u

 α + 1 ⋅ ln 3  2α

⌠  55ii.      ⌡

  

⌠  56i.      ⌡

2 2 u +α 

 

u

3

u2 + α 2  

2 du =

u 8

1 3

u2 + α 2  

u

du =

2 2 u +α

2 α ⋅

2

u

2 α ⋅u

2 2 u −α

du =

⌠  54i.      ⌡

  

⋅ ln

du =

1

α



1

du =

1 3 u ⋅

α

⋅ acos

2 2 u +α

1 2 u ⋅

1

 2 2 ⋅ 2u + 5⋅ α  ⋅

 2 2 3⋅ α u +α + ⋅ lnu + 8 4

2 2 u +α 

⌠  56ii.      ⌡

1 3

u2 − α 2  

3

u2 + α 2  

3

3

3

u2 − α 2  

du =

2

−1

−1 2 2 u −α

du =

−u



+ lnu +

2 u

2 2 u +α

2

2 u

2 2 u −α

2 2 u +α

2

2 u

u2 + α 2  

du =

2

u

u2 − α 2  

2 α ⋅

2

u

−u

du =

du =

−u



+ lnu +

2 u

2 2 u −α

3

u2 − α 2  

2 du =

Integral Table Master.mcd

u 8

 2 2 ⋅ 2u − 5⋅ α  ⋅

4  2 2 3⋅ α u −α + ⋅ lnu + 8

2 2 u −α 

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Integral Table

2 2 α −u

Forms Involving

⌠  15.    ⌡ ⌠  57.   ⌡

1

61.

⌠  62.    ⌡ ⌠   63.    ⌡

⌠  65.    ⌡

2

 u 2 α 2 α −u + ⋅ asin  α 2

u 2 2 α − u du = ⋅ 2

1 u⋅

du =

2 α −u 2

1 2 u ⋅

1 α

2 2 α −u 



α

 

2 2 α −u

−

du =

 α−

⋅ ln

2 α ⋅u

2 2 α −u

3 −1  2 2 2 α − u du = ⋅ α − u  3 2

⌠  2 59.  u ⋅  ⌡

⌠      ⌡

 u  α

du = asin

2 2 α −u

⌠  58.  u⋅  ⌡

⌠   60.   ⌡

⌠  64.    ⌡

u  2 2 2 2 α − u du = ⋅ 2⋅ u − α  ⋅ 8

2 α −u

2 2 α −u 2 u

u

du =

du =

4 2 α  u 2 α −u + ⋅ asin  8 α

 α + 2 2 α − u − α ⋅ ln 

2

u

⌠  66.    ⌡

2

2 2 α −u

−

u

du = −

2 α −u  2

u

 

⌠  68.      ⌡

 u  α

− asin

⌠  69.      ⌡

2 2 α −u

2 2 α −u

2 u

du =

2 2 α −u

Integral Table Master.mcd

−u 2

⋅

⌠    67.  ⌡

⌠  70.      ⌡

2 2 α  u α −u + ⋅ asin  2 α 2

Copyright 2009

1 3 u ⋅

2 2 α −u

−

du =

2 2 2⋅ α ⋅ u

2 2 α −u

+

 α ⋅ ln 3  2⋅ α 1

3

α 2 − u2  

2 du =

1 3

α 2 − u2  

3

α 2 − u2  

3

α 2 − u2  

2

2  2 ⋅ 5⋅ α − 2u  ⋅

du =

2 2 α −u

u 2 2 α −u

2 α ⋅

1 2 2 α −u

2

2 u

8

du =

2

u

u

du =

u 2 2 α −u

 u  α

− asin

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Integral Table

2 Forms involving au + bu + c , a ≠ 0

2 2 R = au + bu + c , D = b − 4ac ⌠  71.   ⌡ ⌠  72.   ⌡ ⌠  73.   ⌡ ⌠  74.   ⌡ ⌠  75.   ⌡ ⌠  76.  ⌡

1 R

1 R

du =

du =

 2⋅ a⋅ u + b − ⋅ ln D  2⋅ a⋅ u + b +

2 −D

du =

R

1

D>0

D 

 2⋅ a⋅ u + b   −D  

⋅ atan 

⌠ u 1 b  du = ⋅ ln( R) − ⋅ R 2a 2a  ⌡

1

D 

1

1 R

⌠  78.   ⌡

D<0

⌠  79.   ⌡

du

(

1

⌠  77.   ⌡

⋅ ln 2⋅ a⋅ u + b + 2⋅

a⋅

)

⌠  80.   ⌡

a> 0

R

a

du =

R

R du =

 2⋅ a⋅ u + b  ⋅ asin  −a D  

−1

( 2⋅ a⋅ u + b) ⋅

Integral Table Master.mcd

4a

⌠ R D  − ⋅ 8⋅ a   ⌡

⌠  81.   ⌡

a < 0,D > 0

1

u

du =

R

1 u⋅

du = R

1 u⋅

R

R⋅

du = R

u R⋅

du = R

a

−1

  

1 −c

1

du

R

R+

⋅ ln

c

du =

1

⌠ b  − ⋅ 2⋅ a   ⌡

R

c

u

du

Copyright 2009

2⋅

 c 

 b⋅ u + 2⋅c    u⋅ D 

⋅ asin

− 2⋅ ( 2⋅ a⋅ u + b) D⋅

R

2⋅ ( b⋅ u + 2c ) D⋅

R

⌠  3  2⋅ a⋅ u + b  3D  2 82.  R du = ⋅ R − ⋅  8 ⋅ a 8⋅ a   ⌡

R

b

+

2 ⌠  R+ ⋅ 2  128a  ⌡ 3⋅ D

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Integral Table

Forms Involving trigonometric functions

⌠ u sin( u) ⋅ cos ( u)  2 83.  sin( u) du = − 2 2 ⌡

⌠  93.  u⋅ cos ( u) du = cos ( u) + u⋅ sin( u) ⌡

⌠ u sin( u) ⋅ cos ( u)  2 84.  cos ( u) du = + 2 2 ⌡

⌠  2 2  94.  u ⋅ sin( u) du = 2⋅ u⋅ sin( u) − u − 2 ⋅ cos ( u) ⌡

⌠ 3 cos ( u)  3 85.  sin( u) du = − cos ( u) 3 ⌡

⌠  2 2  95.  u ⋅ cos ( u) du = 2⋅ u⋅ cos ( u) + u − 2 ⋅ sin( u) ⌡

⌠ 3 sin( u)  3 86.  cos ( u) du = sin( u) − 3 ⌡

⌠ ⌠ n− 1 − sin( u) ⋅ cos ( u) n− 1   n 96.  sin( u) du = + ⋅  sin n n ⌡ ⌡

⌠ u 1  2 2 87.  sin( α ⋅ u) ⋅ cos ( α ⋅ u) du = − ⋅ sin( 4⋅ α ⋅ u) 8 32 ⋅α ⌡

⌠ ⌠ n− 1 cos ( u) ⋅ sin( u) n− 1   n 97.  cos ( u) du = + ⋅  cos n n ⌡ ⌡

⌠  2 88.  tan( u) du = tan( u) − u ⌡

⌠ n− 1 ⌠ tan( u)  n  n− 2 98.  tan( u) du = −  tan( u) du n− 1 ⌡ ⌡

⌠  2 89.  cot( u) du = − cot( u) − u ⌡

⌠ n− 1 ⌠ − cot( u)  n  n− 2 99.  cot( u) du = −  cot( n) du n− 1 ⌡ ⌡

⌠ 1 1  3 90.  sec ( u) du = ⋅ sec ( u) ⋅ tan( u) + ⋅ ln sec ( u) + tan( u) 2 2 ⌡

)

⌠ −1 1  3 91.  csc( u) du = ⋅ csc( u) ⋅ cot( u) + ⋅ ln csc( u) − cot( u) 2 2 ⌡

)

(

(

⌠ ⌠ n− 2 tan( u) ⋅ sec ( u) n− 2   n 100.  sec ( u) du = + ⋅  sec n− 1 n− 1 ⌡ ⌡ ⌠ ⌠ n− 2 − cot( u) ⋅ csc( u) n− 2   n 101.  csc( u) du = + ⋅ n− 1 n− 1 ⌡ ⌡

⌠  92.  u⋅ sin( u) du = sin( u) − u⋅ cos ( u) ⌡

Integral Table Master.mcd

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Integral Table

⌠  m n 102.  cos ( u) ⋅ sin( u) du = ⌡

cos ( u)

m− 1

⋅ sin( u)

n+ 1

m+n

− sin( u)

n− 1

⋅ cos ( u)

⌠ m−1  m− 2 n + ⋅  cos ( u) ⋅ sin( u) du m+n ⌡

m+ 1 +

m+n

− sin( u)

n+ 1

⋅ cos ( u)

m+ 1

m+1

sin( u)

n+ 1

⋅ cos ( u)

n+ 1

m+1 +

⌠ n− 1  m n− 2 ⋅  cos ( u) ⋅ sin( u) du m+n ⌡

⌠ m+ n+ 2  m n− 2 + ⋅  cos ( u) ⋅ sin( u) du m+1 ⌡ ⌠ m+ n+ 2  m n+ 2 ⋅  cos ( u) ⋅ sin( u) du n+ 1 ⌡

2 2 Assume that a ≠ b for #103 -- 104

⌠  108.   ⌡

⌠  109.   ⌡

⌠  110A.    ⌡

1 p + q⋅ sin( α ⋅ u)

1 p + q⋅ sin( α ⋅ u)

  p⋅ tan  ⋅ atan   2 2 p −q  2

du = α⋅

du = α⋅

 α  p⋅ tan 1  ⋅ ln α 2 2  q −p  p⋅ tan  

1 2 2 2 2 p ⋅ sin( α ⋅ u) + q ⋅ cos ( α ⋅ u)

du =

1 α ⋅ p⋅ q

⋅ atan

⌠ 1 sin[ ( − α + β ) ⋅ u] 1 sin[ ( α + β ) ⋅ u]  103.  sin( α ⋅ u) ⋅ sin( β ⋅ u) du → ⋅ − ⋅ 2 ( − α + β ) 2 (α + β) ⌡ ⌠ − 1 cos [ ( α + β ) ⋅ u] 1 cos [ ( − α + β ) ⋅ u]  104.  sin( α ⋅ u) ⋅ cos ( β ⋅ u) du → ⋅ + ⋅ 2 (α + β ) 2 ( −α + β ) ⌡ ⌠ 1 sin[ ( − α + β ) ⋅ u] 1 sin[ ( α + β ) ⋅ u]  105.  cos ( α ⋅ u) ⋅ cos ( β ⋅ u) du → ⋅ + ⋅ 2 ( − α + β ) 2 (α + β ) ⌡

⌠  106A.   ⌡

⌠  107A.   ⌡

1 p + q⋅ cos ( α ⋅ u)

1 p + q⋅ cos ( α ⋅ u)

Integral Table Master.mcd

  α⋅u  tan    2  ⋅ atan ( p − q) ⋅  2 2 2 2 p −q p −q  

2 2 p >q

  α⋅u  tan  1  2  ⋅ ln 2 2   α⋅u q −p  tan    2 

2 2 q >p

2

du = α⋅

du = α⋅

+

−

q+ p



q− p q+ p



q− p

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Integral Table

Form involving exponential functions

⌠ α ⋅u e  α ⋅u 111.  u⋅ e du = ⋅ ( α ⋅ u − 1) 2 ⌡ α

112.

⌠ α⋅u e  α ⋅u 113.  e ⋅ sin( β ⋅ u) du = ⋅ ( α ⋅ sin( β ⋅ u) − 2 2 ⌡ α +β

(

)

⌠ α ⋅u e  α ⋅u 114.  e ⋅ cos ( β ⋅ u) du = ⋅ ( α ⋅ cos ( β ⋅ u) + 2 2 ⌡ α +β

⌠ α ⋅u e  2 α⋅u  2 2  u ⋅ e d u = ⋅ α ⋅ u − 2⋅ α ⋅ u + 2  3 ⌡ α

(

)

Forms involving inverse trigonometric functions α > 0 ⌠   u  u 115.  asin  du = u⋅ asin  + α   α  ⌡ ⌠   u  u 116.  acos  du = u⋅ acos  − α   α  ⌡

⌠ 2 2  2⋅ u − α  u  u u 118.  u⋅ asin  du = ⋅ asin  + ⋅ α   α 4 4  ⌡

2 α −u 2

⌠ 2 2  2⋅ u − α  u  u u 119.  acos  du = ⋅ acos  − ⋅ α   4 α 4  ⌡

2 α −u 2

⌠  1  2  u  u  u⋅ 2 117.  u⋅ atan   du = ⋅ u + α  ⋅ atan   − α 2   α 2  ⌡

⌠   u  u  α  2 2 117.  atan   du = u⋅ atan   − ⋅ lnα + u  α   α 2  ⌡

Integral Table Master.mcd

a

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