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  4. The value of the integral ∫ limits(π/2)-(π/2) sin4 x (1+ log((2+ sin x/2- sin x)))dx is :

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Q. The value of the integral $\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \sin^{4} x \left(1+\log\left(\frac{2+\sin x}{2-\sin x}\right)\right)dx $ is :

JEE MainJEE Main 2018Integrals

Solution:

$I=\int\limits_{-\pi/ 2}^{\pi/ 2} sin^{4}x\left(1+log \left(\frac{2+sin\,x}{2-sin\,x}\right)\right)dx ..........\left(i\right)$
Use proerties $\int\limits^{b}_{0} f\left(x\right)dx=\int^{b}_{a} f\left(a+b-x\right)dx$
$=\int\limits_{-\pi/ 2}^{\pi/ 2}\,sin^{4}\,x\left(1+log\left(\frac{2-sin\,x}{2+sin\,x}\right)\right)dx ............\left(ii\right)$
by (i) + (ii)
$\Rightarrow 2I=\int_{-\pi/ 2}^{\pi/ 2}\,2\,sin^{4}\,xdx$
$\Rightarrow I=2 \int_{-\pi/ 2}^{\pi/ 2}sin^{4}\,xdx=$
$=x. \frac{3.1}{4.2}. \frac{\pi}{2}=\frac{3\pi}{8}$