Question

Solve $\int \frac{1}{\tan x+\cos x+\sec x+\csc x}dx.$

Answer 1

Let $I=\int \frac{1}{\tan x+\cot x+\sec x+\csc x}dx$

$I=\int \frac{1}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}+\frac{1}{\cos x}+\frac{1}{\sin x}}dx$

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

$=\int \frac{\sin x\cos x}{1+\cos x+\sin x}dx$

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

$=\int \frac{2\cos \frac{x}{2}\left(\sin \frac{x}{2}\cos x\right)}{2\cos \frac{x}{2}\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)}dx$

$=\int \frac{\sin \frac{x}{2}\left({\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}\right)}{\cos \frac{x}{2}+\sin \frac{x}{2}}dx$

$=\int \frac{\sin \frac{x}{2}\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)}{\cos \frac{x}{2}+\sin \frac{x}{2}}dx$

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

$=\frac{1}{2}\int 2\sin \frac{x}{2}\cos \frac{x}{2}dx-\int \frac{1-\cos x}{2}dx$

$=\frac{1}{2}\int \sin xdx-\frac{1}{2}\int dx+\frac{1}{2}\int \cos xdx$

$=-\frac{1}{2}\cos x-\frac{1}{2}x+\frac{1}{2}\sin x+c$

$=\frac{\sin x-\cos x-x}{2}+c$