Integrals of Logarithmic Functions

List of integrals involving logarithmic functions

\( \int \ln(cx)dx = x\ln(cx) – x \)

 

\( \int \ln(ax+b)dx = x\ln(ax+b) – x + \frac{b}{a}\ln(ax + b) \)

 

\( \int (\ln x)^2dx = x(\ln x)^2 – 2x\ln x + 2x \)

 

\( \int (\ln (cx))^ndx = x(\ln x)^n – n\cdot\int (\ln (cx))^{n-1}dx \)

 

\( \int \frac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\frac{(\ln x)^i}{i\cdot i!} \)

 

\( \int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1} \int \frac{dx}{(\ln x)^{n-1}} \)

 

\( \int x^m \cdot \ln xdx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2} \right) \quad ( \text{fot } m\ne1) \)

 

\( \int x^m \cdot (\ln x)^ndx = \frac{x^{m+1}(\ln x)^n}{m+1} – \frac{n}{m+1}\int x^m(\ln x)^{n-1}dx \quad (\text{for } m \ne 1) \)

 

\( \int \frac{(\ln x)^n}{x}dx = \frac{(\ln x)^{n+1}}{n+1}, \quad(\text{for } n\ne 1) \)

 

\( \int \frac{\ln x^n}{x}dx = \frac{\left(\ln x^n \right)^2}{2n},\quad (\text{for } n \ne 0 ) \)

 

\( \int \frac{\ln x}{x^m}dx = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2x^{m-1}},\quad(\text{fot }m\ne1) \)

 

\( \int \frac{(\ln x)^n}{x^m}dx = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1}}{x^m}dx,\quad(\text{fot }m\ne1) \)

 

\( \int \frac{dx}{x\cdot \ln x} = \ln|\ln x| \)

 

\( \int \frac{dx}{x^n\cdot \ln x} = \ln|\ln x| + \sum\limits_{i=1}^\infty(-1)^i \frac{(n-1)^i(\ln x)^i}{i\cdot i!} \)

 

\( \int \frac{dx}{x(\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}},\quad(\text{for }n\ne 1) \)

 

\( \int \ln(x^2 + a^2)dx = x\,\ln(x^2 + a^2) – 2x + 2a\,\arctan\frac{x}{a} \)

 

\( \int \sin(\ln x)dx = \frac{x}{2}(\sin(\ln x)-\cos(\ln x)) \)

 

\( \int \cos(\ln x)dx = \frac{x}{2}(\sin(\ln x) + \cos(\ln x)) \)

 

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