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

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

Answers (1)

best_answer

Answer:

\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}

 

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