Question

What is the integration of $\ln x$?

Answer 1

Find integration of $\ln x$.

From integration by parts:

$\mathbf{\int }\mathit{u}\mathit{v}\mathbf{}\mathit{d}\mathit{x}\mathbf{=}\mathit{u}\mathbf{\int }\mathit{v}\mathbf{}\mathit{d}\mathit{x}\mathbf{-}\mathbf{\int }\left(\int vdx\right)\frac{\mathbf{d}\mathbf{u}}{\mathbf{d}\mathbf{x}}\mathit{d}\mathit{x}$

Let $u=\ln x,v=1$

To find$\int \ln xdx$, Put $u=\ln x,v=1$ in $\int uvdx=u\int vdx-\int \left(\int vdx\right)\frac{du}{dx}dx$:

$\begin{array}{rcl}\int \ln xdx& =& \ln x\left(x\right)-\int x\left(\frac{1}{x}\right)dx\left\{\because \frac{d}{dx}\left(\ln x\right)=\frac{1}{x}\right\}\\ & \Rightarrow & \int \ln xdx=x\ln x-\int dx\\ & \Rightarrow & \int \ln xdx=x\ln x-x+C\left\{\because \int dx=x.\right\}\end{array}$

Hence, the required integral is $\begin{array}{rcl}\int \ln xdx& =& x\ln x-x+C\end{array}$ where $C$ is a constant of integration .