Question

Evaluate $\int \frac{\log x}{x^2}dx$.

Answer 1

$\int \frac{\log xdx}{x^2}$

Integrating by parts,

$\int u.vdx=u\int vdx$ $-\int (\frac{du}{dx}\int vdx)dx$

Let $u=\log x\Rightarrow du=\frac{1}{x}dx$

and $v=\frac{1}{x^2}$

$\int \frac{\log xdx}{x^2}=\log x\int \frac{1}{x^2}dx$ $-\int (\frac{d\log x}{dx}\int \frac{1}{x^2}dx)dx$

$=\frac{-\log x}{x}-\int \frac{-1}{x}\times \frac{1}{x}dx$

$=\frac{-\log x}{x}+\int \frac{1}{x^2}dx$

$=\frac{-\log x}{x}-\frac{1}{x}$

$=\frac{-1-\log x}{x}+c$

where $c$ is the constant of integration.