#### Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Very short answer type question 18

Hint: You must know the integration rules of trigonometric function with its limits

Given: $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta$

Solution:  $I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta$            .............(i)

Let  $f(x)=\log \left(\frac{a-\sin \theta}{a+\sin \theta}\right)$

$f(-x)=\log \left(\frac{a-\sin (-\theta)}{a+\sin (-\theta)}\right)$

$=\log \left(\frac{a+\sin \theta}{a-\sin \theta}\right) \quad[\because \sin (-x)=-\sin x]$

\begin{aligned} &=-\log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) \\ &=-\mathrm{f}(\mathrm{x}) \end{aligned}

Hence, f(x) is an odd function.

Since, $\int_{-a}^{a} f(x) d x=0$ if f(x) is an odd.

$\therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{a-\sin \theta}{a+\sin \theta}\right) d \theta=0$