#### Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 10

$\frac{\pi}{2}-1$

Given:

$\int_{0}^{1} \sqrt{\frac{1-x}{1+x}} d x$

Hint:

Explanation:

Let

\begin{aligned} &I=\int_{0}^{1} \sqrt{\frac{1-x}{1+x}} d x \\\\ &=\int_{0}^{1} \sqrt{\frac{1-x}{1+x} \times \frac{1-x}{1-x}} d x \end{aligned}

\begin{aligned} &=\int_{0}^{1} \sqrt{\frac{(1-x)^{2}}{1-x^{2}}} d x \\\\ &=\int_{0}^{1} \frac{1-x}{\sqrt{1-x^{2}}} d x \end{aligned}

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

\begin{aligned} &=\left[\sin ^{-1} x\right]_{0}^{1}+\frac{1}{2}\left[\frac{\left(1-x^{2}\right)^{1 / 2}}{1 / 2}\right]_{0}^{1} \\\\ &=\left[\sin ^{-1}(1)-\sin ^{-1}(0)\right]+\frac{1}{2}\left[0-\frac{1}{1 / 2}\right] \end{aligned}

\begin{aligned} &=\left(\frac{\pi}{2}-0\right)+\frac{1}{2}(-2) \\\\ &=\frac{\pi}{2}-1 \end{aligned}