List of integrals of logarithmic functions explained

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

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

Integrals involving only logarithmic functions

\int log _{a} xdx = x log _{a} x - (x)/(ln a) = (x)/(ln a) ( ln x - 1 ) \int ln (ax)dx = x ln (ax) - x = x( ln (ax) - 1 ) \int ln (ax + b)dx = (ax + b)/(a) ( ln (ax + b) - 1 ) \int ( ln x) ^{2dx} = x( ln x) ^{2} - 2 x ln x + 2 x \int ( ln x) ^{ndx} = ( -1 ) ^{n} n! x \sum ^{n} _{k =0} (( -ln x) ^{k})/(k!) \int (dx)/(ln x) = ln | ln x| + ln x + \sum ^{inf ty} _{k =2} (( ln x) ^{k})/(k ⋅ k!) \int (dx)/(ln x) = \operatorname{li}(x) , the logarithmic integral.\int (dx)/(( ln x) ^{n}) =- (x)/((n -1 )( ln x) ^{n-1}) + (1)/(\int n -1) (dx)/(( ln x) ^{n-1}) (for n ≠ 1 ) \int ln f(x)dx = x ln f(x) - \int x (f'(x))/(f(x)) dx (fordifferentiable f(x) > 0 )

Integrals involving logarithmic and power functions

\int x ^{mln} xdx = x ^{m+1} \left( (ln x)/(- m +1) (1)/((m +1 ) ^{2}) \right) (for m ≠ -1 ) \int x ^{m} ( ln x) ^{ndx} = (x ^{m+1} ( ln x) ^{n})/(m +1) - (n)/(m +1) \int x ^{m} ( ln x) ^{n-1} dx (for m ≠ -1 ) \int (( ln x) ^{ndx})/(x) = (( ln x) ^{n+1})/(n +1) (for n ≠ -1 ) \int (ln xdx)/(x ^{m}) =- (ln x)/(- (m -1 )x ^{m-1}) (1)/((m -1 ) ^{2} x ^{m-1}) (for m ≠ 1 ) \int (( ln x) ^{ndx})/(x ^{m}) =- (( ln x) ^{n})/((m -1 )x ^{m-1}) + (n)/(\int m -1) (( ln x) ^{n-1} dx)/(x ^{m}) (for m ≠ 1 ) \int (x ^{mdx})/(( ln x) ^{n}) =- (x ^{m+1})/((n -1 )( ln x) ^{n-1}) + (m +1)/(\int n -1) (x ^{m} dx)/(( ln x) ^{n-1}) (for n ≠ 1 ) \int (dx)/(x ln x) = ln \left| ln x\right| \int (dx)/(x ln x lnln x) = ln \left| ln \left| ln x\right| \right| , etc.\int (dx)/(x lnln x) = \operatorname{li}( ln x) \int (dx)/(x ^{nln} x) = ln \left| ln x\right| + \sum ^{inf ty} _{k =1} ( -1 ) ^{k ((n -1 ) ^{k(ln} x) ^{k})/(k ⋅ k!)} \int (dx)/(x( ln x) ^{n}) =- (1)/((n -1 )( ln x) ^{n-1}) (for n ≠ 1 ) \int ln (x ^{2+a} ^{2)dx} = x ln (x ^{2+a} ^{2)-2x+2a\tan} ^{-1} (x)/(a) \int (x)/(x ^{2+a} ^{2}) ln (x ^{2+a} ^{2)dx} = (1)/(4) ln ^{2(x} ^{2+a} ^{2)}

Integrals involving logarithmic and trigonometric functions

\int \sin ( ln x)dx = (x)/(2) (\sin ( ln x) - \cos ( ln x)) \int \cos ( ln x)dx = (x)/(2) (\sin ( ln x) + \cos ( ln x))

Integrals involving logarithmic and exponential functions

\int e ^{x} \left(x ln x - x - (1)/(x) \right)dx = e ^{x} (x ln x - x - ln x) \int (1)/(e ^{x}) \left( (1)/(x) -ln x \right)dx = (ln x)/(e ^{x}) \int e ^{x} \left( (1)/(ln x) - (1)/(x( ln x) ^{2}) \right)dx = (e ^{x})/(ln x)

n consecutive integrations

For

n consecutive integrations, the formula\int ln xdx = x( ln x - 1 ) + C _{0}

generalizes to

\int ... i\int ln xdx ... m dx = (x ^{n})/(n!) \left( ln x - \sum ^{n} _{k =1} (1)/(k) \right) + \sum ^{n -1} _{k =0} C _{k} (x ^{k})/(k!)

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on page 69.