$\int \ln(cx)dx = x\ln(cx) - x + C  \\$ $\int \ln(ax+b)dx = x\ln(ax+b) - x + \dfrac{b}{a}~\ln(ax + b) + C \\$ $\int (\ln x)^2dx = x(\ln x)^2 - 2x\ln x + 2x + C \\$ $\int (\ln (cx))^ndx = x(\ln x)^n - n\cdot\int (\ln (cx))^{n-1}dx + C \\$ $\int \dfrac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\dfrac{(\ln x)^i}{i\cdot i!} + C \\$ $\int \dfrac{dx}{(\ln x)^n} = -\dfrac{x}{(n-1)(\ln x)^{n-1}} + \dfrac{1}{n-1} \int \dfrac{dx}{(\ln x)^{n-1}} + C \\$ $\int x^m \cdot \ln x~dx = x^{m+1}\left(\dfrac{\ln x}{m+1}-\dfrac{1}{(m+1)^2} \right)+C \quad ( \text{for } m\ne1)\\$ $\int x^m \cdot (\ln x)^n~dx = \dfrac{x^{m+1}(\ln x)^n}{m+1} - \dfrac{n}{m+1}\int x^m(\ln x)^{n-1}dx+C \quad (\text{for } m \ne 1)  \\$ $\int \dfrac{(\ln x)^n}{x}~dx = \dfrac{(\ln x)^{n+1}}{n+1}+ C, \quad(\text{for } n\ne 1)  \\$ $\int \dfrac{\ln x^n}{x}~dx = \dfrac{\left(\ln x^n \right)^2}{2n}+ C,\quad (\text{for } n \ne 0 )\\$ $\int \dfrac{(\ln x)^n}{x^m}~dx = -\dfrac{(\ln x)^n}{(m-1)x^{m-1}} + \dfrac{n}{m-1}\int\dfrac{(\ln x)^{n-1}}{x^m}~dx+ C ,\quad(\text{for }m\ne1)  \\$ $\int \dfrac{dx}{x\cdot \ln x} = \ln|\ln x| + C \\$ $\int \dfrac{dx}{x^n\cdot \ln x} = \ln|\ln x| + \sum\limits_{i=1}^\infty(-1)^i \dfrac{(n-1)^i(\ln x)^i}{i\cdot i!} + C \\$ $\int \dfrac{dx}{x(\ln x)^n} = -\dfrac{1}{(n-1)(\ln x)^{n-1}} + C ,\quad(\text{for }n\ne 1) \\$ $\int \ln(x^2 + a^2)dx = x\,\ln(x^2 + a^2) - 2x + 2a\,\tan^{-1}\dfrac{x}{a} + C \\$ $\int \sin(\ln x)dx = \dfrac{x}{2}(\sin(\ln x)-\cos(\ln x)) + C \\$ $\int \cos(\ln x)dx = \dfrac{x}{2}(\sin(\ln x) + \cos(\ln x)) + C $

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