Question

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

Answer 1

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

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

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

$=\frac{1}{2}\int x{\sec }^{2}x/2dx+\int \tan x/2dx$

$=\frac{1}{2}[x\cdot \int {\sec }^{2}x/2dx-\int \left(\frac{d}{dx}\right(x)\cdot \int {\sec }^{2}x/2dx)dx]+\int \tan x/2dx$

$=\frac{1}{2}[x\cdot \frac{\tan x/2}{1/2}-\int \frac{\tan x/2dx}{1/2}]+\int \tan x/2dx+c$

$=x\tan x/2-\int \tan x/2dx+\int \tan x/2dx+c$

$=x\tan x/2+c$