Question

$\int \frac{dx}{\tan x+\cot x+\sec x+\cos ecx}$

Answer 1

We have,

$\int \frac{dx}{\tan x+\mathrm{cotx}+secx+cscx}$

$\Rightarrow \int \frac{dx}{\frac{\sin x}{\cos x}+\frac{\mathrm{cosx}}{\sin x}+\frac{1}{\cos x}+\frac{1}{\sin x}}$

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

$\Rightarrow \int \frac{\sin x\cos xdx}{1+\sin x+\cos x}$

$\Rightarrow \int \frac{2\sin \frac{x}{2}\cos \frac{x}{2}\cos xdx}{1+2{\cos }^2\frac{x}{2}-1+2\sin \frac{x}{2}\cos \frac{x}{2}}$

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

$\Rightarrow \int \frac{2\cos \frac{x}{2}\sin \frac{x}{2}(\cos \frac{x}{2}-\sin \frac{x}{2})(\cos \frac{x}{2}+\sin \frac{x}{2})}{2\cos \frac{x}{2}(\cos \frac{x}{2}+\sin \frac{x}{2})}dx$

$\Rightarrow \int \frac{\sin \frac{x}{2}(\cos \frac{x}{2}-\sin \frac{x}{2})}{1}dx$

$\Rightarrow \int \sin \frac{x}{2}(\cos \frac{x}{2}-\sin \frac{x}{2})dx$

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

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

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

On integration and we get,

$\Rightarrow \frac{1}{2}(-\cos x)-\frac{1}{2}x+\frac{1}{2}\sin x+C$

$\Rightarrow \frac{1}{2}\sin x-\frac{1}{2}\cos x-\frac{1}{2}x+C$

Hence, this is the answer.