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Math formulas: Integrals of logarithmic functions

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List of integrals involving logarithmic functions

ln(cx)dx=xln(cx)x \int \ln(cx)dx = x\ln(cx) - x
ln(ax+b)dx=xln(ax+b)x+baln(ax+b) \int \ln(ax+b)dx = x\ln(ax+b) - x + \frac{b}{a}\ln(ax + b)
(lnx)2dx=x(lnx)22xlnx+2x \int (\ln x)^2dx = x(\ln x)^2 - 2x\ln x + 2x
(ln(cx))ndx=x(lnx)nn(ln(cx))n1dx \int (\ln (cx))^ndx = x(\ln x)^n - n\cdot\int (\ln (cx))^{n-1}dx
dxlnx=lnlnx+lnx+n=2(lnx)iii! \int \frac{dx}{\ln x} = \ln|\ln x|+\ln x+\sum\limits_{n=2}^\infty\frac{(\ln x)^i}{i\cdot i!}
dx(lnx)n=x(n1)(lnx)n1+1n1dx(lnx)n1 \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}}
xmlnxdx=xm+1(lnxm+11(m+1)2)(fot m1) \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)
xm(lnx)ndx=xm+1(lnx)nm+1nm+1xm(lnx)n1dx(for m1) \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)
(lnx)nxdx=(lnx)n+1n+1,(for n1) \int \frac{(\ln x)^n}{x}dx = \frac{(\ln x)^{n+1}}{n+1}, \quad(\text{for } n\ne 1)
lnxnxdx=(lnxn)22n,(for n0) \int \frac{\ln x^n}{x}dx = \frac{\left(\ln x^n \right)^2}{2n},\quad (\text{for } n \ne 0 )
lnxxmdx=lnx(m1)xm11(m1)2xm1,(fot m1) \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)
(lnx)nxmdx=(lnx)n(m1)xm1+nm1(lnx)n1xmdx,(fot m1) \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)
dxxlnx=lnlnx \int \frac{dx}{x\cdot \ln x} = \ln|\ln x|
dxxnlnx=lnlnx+i=1(1)i(n1)i(lnx)iii! \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!}
dxx(lnx)n=1(n1)(lnx)n1,(for n1) \int \frac{dx}{x(\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}},\quad(\text{for }n\ne 1)
ln(x2+a2)dx=xln(x2+a2)2x+2aarctanxa \int \ln(x^2 + a^2)dx = x\,\ln(x^2 + a^2) - 2x + 2a\,\arctan\frac{x}{a}
sin(lnx)dx=x2(sin(lnx)cos(lnx)) \int \sin(\ln x)dx = \frac{x}{2}(\sin(\ln x)-\cos(\ln x))
cos(lnx)dx=x2(sin(lnx)+cos(lnx)) \int \cos(\ln x)dx = \frac{x}{2}(\sin(\ln x) + \cos(\ln x))

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