Question

Solve $\int \frac{cos4\theta +1}{cot\theta -tan\theta }d\theta$

Answer 1

$\int \frac{\cos 4\theta +1}{\cot \theta -\tan \theta }d\theta$

$1+{\cos }^2\theta =2{\cos }^2\theta$

$\int \frac{{\cos }^{2}2\theta }{\frac{\cos \theta }{\sin \theta }-\frac{\sin \theta }{\cos \theta }}d\theta$

$\int \frac{2{\cos }^2\theta }{\frac{{\cos }^2\theta -{\sin }^2\theta }{\sin \theta \cos \theta }}d\theta$

$\int \frac{2{\cos }^{2}2\theta }{{\cos }^2\theta }\left(\sin \theta \cos \theta \right)d\theta$

$\cos 2\theta \left(2\sin \theta \cos \theta \right)d\theta$

$\int \cos 2\theta \sin 2\theta d\theta$

$\frac{1}{2}\int 2\cos 2\theta \sin 2\theta d\theta$

$\frac{1}{2}\int \sin 4\theta d\theta$

Let $4\theta =x$

$4d\theta =dx$

$d\theta =\frac{1}{4}dx$

$\frac{1}{8}\int \sin xdx$

$\frac{1}{8}(-\cos x)+C$

$\frac{-1}{8}\cos 4\theta +C$(Ans)