Question

Evaluate : $\int \frac{{\sin }^{2}x-{\cos }^{2}x}{\sin x\cos x}dx.$

Answer 1

We have,

$I=\int \frac{{\sin }^{2}x-{\cos }^{2}x}{\sin x\cos x}dx$

$I=-2\int \frac{{\cos }^{2}x-{\sin }^{2}x}{2\sin x\cos x}dx$

$I=-2{\int }^{}\frac{\cos 2x}{\sin 2x}dx=-2{\int }^{}\cot 2xdx=\frac{-2}{2}\ln \left|\sin 2x\right|+c=-\ln \left|\sin 2x\right|+C$

Hence, this is the answer.