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List of integrals of logarithmic functions

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The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals.

Contents

Note: x>0 is assumed throughout this article, and the constant of integration is omitted for simplicity.

Integrals involving only logarithmic functions

log a x d x = x ln x x ln a ln a x d x = x ln a x x ln ( a x + b ) d x = ( a x + b ) ln ( a x + b ) a x a ( ln x ) 2 d x = x ( ln x ) 2 2 x ln x + 2 x ( ln x ) n d x = x k = 0 n ( 1 ) n k n ! k ! ( ln x ) k = Θ ( x ( ln x ) n ) d x ln x = ln | ln x | + ln x + k = 2 ( ln x ) k k k ! d x ln x = li ( x ) = the logarithmic integral (asymptotically, li ( x ) = Θ ( x ln x ) ). d x ( ln x ) n = x ( n 1 ) ( ln x ) n 1 + 1 n 1 d x ( ln x ) n 1 (for  n 1 )

Integrals involving logarithmic and power functions

x m ln x d x = x m + 1 ( ln x m + 1 1 ( m + 1 ) 2 ) (for  m 1 ) x m ( ln x ) n d x = x m + 1 ( ln x ) n m + 1 n m + 1 x m ( ln x ) n 1 d x (for  m 1 ) ( ln x ) n d x x = ( ln x ) n + 1 n + 1 (for  n 1 ) ln x n d x x = ( ln x n ) 2 2 n (for  n 0 ) ln x d x x m = ln x ( m 1 ) x m 1 1 ( m 1 ) 2 x m 1 (for  m 1 ) ( ln x ) n d x x m = ( ln x ) n ( m 1 ) x m 1 + n m 1 ( ln x ) n 1 d x x m (for  m 1 ) x m d x ( ln x ) n = x m + 1 ( n 1 ) ( ln x ) n 1 + m + 1 n 1 x m d x ( ln x ) n 1 (for  n 1 ) d x x ln x = ln | ln x | d x x ln x ln ln x = ln | ln | ln x | | , etc. d x x ln ln x = li ( ln x ) where li is the logarithmic integral. d x x n ln x = ln | ln x | + k = 1 ( 1 ) k ( n 1 ) k ( ln x ) k k k ! d x x ( ln x ) n = 1 ( n 1 ) ( ln x ) n 1 (for  n 1 ) ln ( x 2 + a 2 ) d x = x ln ( x 2 + a 2 ) 2 x + 2 a tan 1 x a x x 2 + a 2 ln ( x 2 + a 2 ) d x = 1 4 ln 2 ( x 2 + a 2 )

Integrals involving logarithmic and trigonometric functions

sin ( ln x ) d x = x 2 ( sin ( ln x ) cos ( ln x ) ) cos ( ln x ) d x = x 2 ( sin ( ln x ) + cos ( ln x ) )

Integrals involving logarithmic and exponential functions

e x ( x ln x x 1 x ) d x = e x ( x ln x x ln x ) 1 e x ( 1 x ln x ) d x = ln x e x e x ( 1 ln x 1 x ln 2 x ) d x = e x ln x

n consecutive integrations

For n consecutive integrations, the formula

ln x d x = x ( ln x 1 ) + C 0

generalizes to

ln x d x d x = x n n ! ( ln x k = 1 n 1 k ) + k = 0 n 1 C k x k k !

References

List of integrals of logarithmic functions Wikipedia


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