Question

Find: \(\int \log~x.dx\)

Answer 1

\(\int \log~x.dx\)
\( = \int \overset{\underbrace{1}}{II}.\overset{\underbrace{(\log x)}}{I}.dx\)
Usinf by parts
\(\int (\log~x).1~dx\)
\( = \log~x \int 1.dx - \int \left ( \dfrac{d(\log~x)}{dx} \int 1.dx \right ).dx\)
\( = (\log~x) x - \int \dfrac{1}{x} .x.dx\)
\( = x~\log~x - \int 1.dx = x~\log~x - x+ C\)
Where \(C\) is constant of integration.