Integrate : $\int \limits_0^{\pi/2} \bigg( \large\frac{\theta}{ \sin \theta }\bigg)^2.$$d \theta (a)\;\pi \log 2 \\(b)\;\pi \log 3 \\(c)\;\pi^2 \log 4 \\ (d)\;None 1 Answer \int \limits _0^{\pi/2} \theta ^2 cosec^2 \theta . d \theta => [\theta ^2 . [-(\cot \theta)]_0^{\pi/2} + \int _0^{\pi/2} 2\theta. \cot \theta d\theta => [-\theta ^2 [\cot \theta ]_0^{\pi/2}+ 2 \int _0^{\pi/2} \theta . \cot \theta . d \theta => [\large\frac{-\pi^2}{2}. \cot \large\frac{\pi}{2}-0]+ 2 \int \limits _0^{\pi/2}$$ \theta . \cot \theta d \theta$
=> $+0+2 \int \limits _0^{\pi/2} \theta . \cot \theta d \theta$
=> $2.[\theta . \log \sin \theta ]_0^{\pi/2} - 2 \int \limits _0^{\pi / 2} \log \sin \theta . d \theta$
=> $2.[\theta . \log \sin \theta ]_0^{\pi/2} - 2 \times (\large\frac{-\pi}{2}.$$\log 2) \lim _ {a \to \theta } \theta . \log \sin \theta \lim _{\theta \to 0 } \large\frac{\log \sin \theta }{\theta}$$+\pi \log 2$
$\lim _{\theta \to 0 } \large\frac{{-\theta}^2 }{ \tan \theta} $$+\pi \log 2 => \lim _{\theta \to 0 } \bigg(\large\frac{\theta}{ \tan \theta}\bigg)$$ \theta +\pi \log 2$
$\qquad= \pi \log 2$
Hence a is the correct answer.