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Seznam integrálů logaritmických funkcí

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Toto je seznam integrálů (primitivních funkcí) logaritmických funkcí (v následujících integrálech se předpokládá, že x > 0).


\(\int\ln cx\,\,\mathrm{d}x = x\ln (cx) - x\)
\(\int (\ln x)^2\; \,\mathrm{d}x = x(\ln x)^2 - 2x\ln x + 2x\)
\(\int (\ln cx)^n\; \,\mathrm{d}x = x(\ln cx)^n - n\int (\ln cx)^{n-1} \,\mathrm{d}x\)
\(\int \frac{\,\mathrm{d}x}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!}\)
\(\int \frac{\,\mathrm{d}x}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1}\int\frac{\,\mathrm{d}x}{(\ln x)^{n-1}} \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int x^m\ln x\;\,\mathrm{d}x = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(pro }m\neq -1\mbox{)}\)
\(\int x^m (\ln x)^n\; \,\mathrm{d}x = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} \,\mathrm{d}x \qquad\mbox{(pro }m\neq -1\mbox{)}\)
\(\int \frac{(\ln x)^n\; \,\mathrm{d}x}{x} = \frac{(\ln x)^{n+1}}{n+1} \qquad\mbox{(pro }n\neq -1\mbox{)}\)
\(\int \frac{\ln x\,\,\mathrm{d}x}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(pro }m\neq 1\mbox{)}\)
\(\int \frac{(\ln x)^n\; \,\mathrm{d}x}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} \,\mathrm{d}x}{x^m} \qquad\mbox{(pro }m\neq 1\mbox{)}\)
\(\int \frac{x^m\; \,\mathrm{d}x}{(\ln x)^n} = -\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m \,\mathrm{d}x}{(\ln x)^{n-1}} \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int \frac{\,\mathrm{d}x}{x\ln x} = \ln|\ln x|\)
\(\int \frac{\,\mathrm{d}x}{x^n\ln x} = \ln|\ln x| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!}\)
\(\int \frac{\,\mathrm{d}x}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(pro }n\neq 1\mbox{)}\)
\(\int \sin (\ln x)\;\,\mathrm{d}x = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))\)
\(\int \cos (\ln x)\;\,\mathrm{d}x = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))\)
\(\int e^x (x \ln x - x - \frac{1}{x})\;\,\mathrm{d}x = e^x (x \ln x - x - \ln x) \)

Externí odkazy

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