Question

Evaluate the following :
$\int \left(\frac{x\cos x-\sin x}{x\sin x}\right)dx$

Answer 1

We have,

$\int \frac{x\cos x-\sin x}{x\sin x}dx$

$\Rightarrow \int (\frac{x\cos x}{x\sin x}-\frac{\sin x}{x\sin x})dx$

$\Rightarrow \int (\cot x-x^{-1})dx$

$\Rightarrow \int \cot xdx-\int \frac{1}{x}dx\therefore \int \frac{1}{x}dx=\log x$

$\Rightarrow \log \sin x-\log x+C$

Hence, this is the answer.