The value of the integral _-/2^/2 ^4x [(2+ x2- x)+1]dx is

Question

The value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2} \sin^4x \left[\log\left(\dfrac{2+\sin x}{2-\sin x}\right)+1\right]dx$ is

Accepted Answer

$I = \int_{-\pi/2}^{\pi/2} \sin^4x\left[1+\log\left(\dfrac{2+\sin x}{2-\sin x}ight)ight]dx$ (1)

Replacing $x 	o -x$: $I = \int_{-\pi/2}^{\pi/2} \sin^4x\left[1+\log\left(\dfrac{2-\sin x}{2+\sin x}ight)ight]dx = \int_{-\pi/2}^{\pi/2} \sin^4x\left[1-\log\left(\dfrac{2+\sin x}{2-\sin x}ight)ight]dx$ (2)

Adding (1) and (2): $2I = 2\int_{-\pi/2}^{\pi/2}\sin^4x\,dx = 4\int_0^{\pi/2}\sin^4x\,dx$

$I = 2\int_0^{\pi/2}\sin^4x\,dx = 2	imes\dfrac{3}{2}	imes\dfrac12	imes\dfrac{\pi}{2}	imes\dfrac{2}{2} = \dfrac{3\pi}{8}$ [By Wallis/Gamma formula]

Solutions · 1