The integral √1+2 x( x+ x),dx, 0

Question

The integral $\displaystyle\int \sqrt{1+2\cot x(\cosec x+\cot x)}\,dx$, $0 < x < \dfrac{\pi}{2}$ is equal to: (where C is a constant of integration)

Accepted Answer

Let $I = \int \sqrt{1+2\cot x\,\cosec x + 2\cot^2x}\,dx = \int \sqrt{\dfrac{\sin^2x+2\cos x+2\cos^2x}{\sin^2x}}\,dx$

$= \int \dfrac{\sqrt{1+2\cos x+\cos^2x}}{\sin x}\,dx = \int \dfrac{1+\cos x}{\sin x}\,dx = \int(\cosec x+\cot x)\,dx$

$I = \log|\cosec x - \cot x| + \log|\sin x| + C = \log|1-\cos x| + C = \log\left(2\sin^2\dfrac{x}{2}\right) + C$

$I = \log\sin^2\dfrac{x}{2} + \log 2 + C = 2\log\left|\sin\dfrac{x}{2}\right| + C_1$

Solutions · 1