Integrate log x

We need to find the integration of log x

Solution

\(\int \log x=\int \frac{\ln (x)}{\ln (10)}dx\) \(=\frac{1}{\\ln 10}\int \ln (x)dx\)

Now let us integrate it by parts

\(\int f(x)g'(x)= [f(x)g(x)]\int f'(x)g(x)dx\) \(f(x)=\ln (x), f'(x)=1/x g(x)=x, g'(x)=1\) \(\int log(x)dx= 1/\ln (10)=[x ln (x)- \int dx]\) \(\int log(x)dx= 1/\ln (10)=[x ln (x)- xdx]\) \(\int log(x)dx= 1/\ln (10)=[x ln (x)- 1] + C\)

Answer

In general, \(\int log(x)dx= 1/\ln (10)=[x ln (x)- 1] + C\)

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