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  • Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 33 maths textbook solution

Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 33 maths textbook solution

Answers (1)

best_answer

Answer:

\frac{\pi }{2}

Hint:

To solve this equation we use \int_{a}^{b} \frac{d}{d x} f(x) d x   formula.

Given:

\int_{0}^{1} \frac{d}{d x}\left\{\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\right\} d x

Solution:

Let

\begin{aligned} I &=\int_{0}^{1} \frac{d}{d x}\left\{\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\right\} d x \\\\ &=\int_{0}^{1} \frac{d}{d x}\left(2 \tan ^{-1} x\right) d x \end{aligned}

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

\begin{aligned} &=2\left[\tan ^{-1} x\right]_{0}^{1} \\\\ &=2\left[\tan ^{-1} 1-\tan ^{-1} 0\right] \\\\ &=2\left[\frac{\pi}{4}\right]-0 \\\\ &=\frac{\pi}{2} \end{aligned}

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