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Latest revision as of 16:12, 26 February 2015
Indefinite Integrals with $ \ln x $
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| $ \int\ln x dx=x\ln x-x +C $ |
| $ \int x\ln x dx=\dfrac{x^{2}}{2}(\ln x-\frac{1}{2})+C $ |
| $ \int x^{m}\ln x dx=\dfrac{x^{m+1}}{m+1}(\ln x-\frac{1}{m+1}) +C $ |
| $ \int\dfrac{\ln x}{x} dx=\frac{1}{2}\ln^{2}x +C $ |
| $ \int\dfrac{\ln x}{x^{2}} dx=-\dfrac{\ln x}{x}-\dfrac{1}{x} +C $ |
| $ \int\ln^{2}x dx=x\ln^{2}x-2x\ln x+2x +C $ |
| $ \int\dfrac{\ln^{n}x}{x} dx=\dfrac{\ln^{n+1}x}{n+1} +C $ |
| $ \int\dfrac{dx}{x\ln x}=\ln(\ln x) +C $ |
| $ \int\dfrac{dx}{\ln x}=\ln(\ln x)+\ln x+\dfrac{\ln^{2}x}{2\cdot2!}+\dfrac{\ln^{3}x}{3\cdot3!}+\cdots +C $ |
| $ \int\dfrac{x^{m}dx}{\ln x}=\ln(\ln x)+(m+1)\ln x+\dfrac{(m+1)^{2}\ln^{2}x}{2\cdot2!}+\dfrac{(m+1)^{3}\ln^{3}x}{3\cdot3!}+\cdots +C $ |
| $ \int\ln^{n}x dx=x\ln^{n}x-n\int\ln^{n-1}x dx $ |
| $ \int x^{m}\ln^{n}x dx=\dfrac{x^{m+1}\ln^{n}x}{m+1}-\dfrac{n}{m+1}\int x^{m}\ln^{n-1}x dx +C $ |
| $ \int\ln(x^{2}+a^{2}) dx=x\ln(x^{2}+a^{2})-2x+2a\tan^{-1}\dfrac{x}{a} +C $ |
| $ \int\ln(x^{2}-a^{2}) dx=x\ln(x^{2}-a^{2})-2x+a\ln(\dfrac{x+a}{x-a}) +C $ |
| $ \int x^{m}\ln(x^{2}\pm a^{2}) dx=\dfrac{x^{m}\ln(x^{2}\pm a^{2})}{m+1}-\dfrac{2}{m+1}\int\dfrac{x^{m+2}}{x^{2}\pm a^{2}}dx $ |
