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

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

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