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

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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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