Question

The value of integral ${\int }_0^1\frac{si{n}^{-1}x}{si{n}^{-1}x+co{s}^{-1}\sqrt{2x-x^2}}dx$ is

• A. 1
• B. $\frac{1}{2}$
• C. 2
• D. 0

Answer 1

The correct option is B $\frac{1}{2}$

Let $I={\int }_0^1\frac{si{n}^{-1}x}{si{n}^{-1}x+co{s}^{-1}\sqrt{2x-x^2}}dx$

$⟹I={\int }_0^1\frac{si{n}^{-1}x}{si{n}^{-1}x+si{n}^{-1}(1-x)}dx$ ------(1)

(since, $co{s}^{-1}\sqrt{2x-x^2}=co{s}^{-1}\sqrt{2(1-x)-(1-x{)}^2}=co{s}^{-1}\sqrt{1-x^2}=si{n}^{-1}(1-x)$

$I={\int }_0^1\frac{si{n}^{-1}(1-x)}{si{n}^{-1}(1-x)+si{n}^{-1}x}dx$ ------(2)

$\left(1\right)+\left(2\right)⟹2I={\int }_0^{1}1dx=(1-0)=1$

$⟹I=\frac{1}{2}$