#### Need solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 11

$\frac{\pi}{\sqrt{a^{2}-b^{2}}}$

Given:

$\int_{0}^{\pi} \frac{1}{a+b \cos x} d x$

Hint:

Using $\int \frac{1}{1+x^{2}} d x$
Explanation:

Let

\begin{aligned} &I=\int_{0}^{\pi} \frac{1}{a+b \cos x} d x \\\\ &\cos x=\frac{1-\tan ^{2} x / 2}{1+\tan ^{2} x / 2} \end{aligned}

\begin{aligned} &=\int_{0}^{\pi} \frac{1}{a+b\left(\frac{1-\tan ^{2} x / 2}{1+\tan ^{2} x / 2}\right)} d x \\\\ &=\int_{0}^{\pi} \frac{1+\tan ^{2} x / 2}{2 a+a \tan ^{2} x / 2+b-b \tan ^{2} x / 2} d x \end{aligned}

\begin{aligned} &=\int_{0}^{\pi} \frac{\sec ^{2 } \frac{x}{2}}{(a-b) \tan ^{2} x / 2+(a+b)} d x \\\\ &\text { Put } \tan {x} /_{2}=t \\\\ &\sec ^{2} x / 2 \cdot{ }^{1} /{ }_{2} d x=d t \\\\ &\sec ^{2} x /{2} d x=2 d t \end{aligned}

$=\int_{0}^{\infty} \frac{2 d t}{(a-b) t^{2}+(a+b)}$

$=\frac{2}{(a-b)} \int_{0}^{\infty} \frac{d t}{t^{2}+\left(\sqrt{\frac{a+b}{a-b}}\right)^{2}}$

$=\frac{2}{a-b}\left\{\tan ^{-1}\left[t \sqrt{\frac{a-b}{a+b}}\right]\right\}_{0}^{\infty} \times \frac{1}{\sqrt{\frac{a+b}{a-b}}}$

\begin{aligned} &=\frac{2}{a-b} \frac{\sqrt{a-b}}{\sqrt{a+b}}\left(\tan ^{-1} \infty-\tan ^{-1} 0\right) \\\\ &=\frac{2}{\sqrt{(a+b)(a-b)}} \cdot\left(\frac{\pi}{2}\right) \\\\ &=\frac{\pi}{\sqrt{a^{2}-b^{2}}} \end{aligned}