Question

Integrate $\int \log \left(\mathrm{x}\right)dx$.

Answer 1

Solve the given integral

Use integration by parts,

$\int \mathrm{I}\left(\mathrm{x}\right)\mathrm{II}\left(\mathrm{x}\right)=\mathrm{I}\left(\mathrm{x}\right)\int \mathrm{II}\left(\mathrm{x}\right)\mathrm{d}\left(\mathrm{x}\right)-\int \frac{d\mathrm{I}\left(\mathrm{x}\right)}{d\mathrm{x}}\int \mathrm{II}\left(\mathrm{x}\right)\mathrm{d}\left(\mathrm{x}\right)$

We know that, $\log \left(\mathrm{x}\right)=\ln \left(\mathrm{x}\right)$

So, for$\int 1\times \log \left(\mathrm{x}\right)$, where $\mathrm{I}\left(\mathrm{x}\right)=\log \left(\mathrm{x}\right),\mathrm{II}\left(\mathrm{x}\right)=1$

$$\begin{gathered} \int \log \left(\mathrm{x}\right)=\log \left(\mathrm{x}\right)\int 1\mathrm{dx}-\int \frac{d\log \left(\mathrm{x}\right)}{d\mathrm{x}}\int 1\mathrm{dx}\left[\because \int \mathrm{cdx}=\mathrm{cx},\mathrm{c}=\mathrm{constant}\right] \\ \Rightarrow \int \log \left(\mathrm{x}\right)=\mathrm{xlog}\left(\mathrm{x}\right)-\int \frac{1}{\mathrm{x}}\mathrm{xdx}\left[\because \frac{d\log \left(\mathrm{x}\right)}{d\mathrm{x}}=\frac{1}{\mathrm{x}}\right] \\ \Rightarrow \int \log \left(\mathrm{x}\right)=\mathrm{x}\left(\log \left(\mathrm{x}\right)-1\right)+\mathrm{C} \end{gathered}$$

Hence, $\int \log \left(\mathrm{x}\right)$ is integrated as $\mathrm{x}\left(\log \left(\mathrm{x}\right)-1\right)+\mathrm{C}$.