JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}\), where C is the constant of integration, then \(g\left(\frac{1}{2}\right)\) equals :
- A \(\frac{\pi}{4} \sqrt{\frac{e}{3}}\)
- B \(\frac{\pi}{6} \sqrt{\frac{e}{3}}\)
- C \(\frac{\pi}{4} \sqrt{\frac{e}{2}}\)
- D \(\frac{\pi}{6} \sqrt{\frac{e}{2}}\)
Answer & Solution
Correct Answer
(B) \(\frac{\pi}{6} \sqrt{\frac{e}{3}}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \frac{d}{d x}\left(\frac{x \cdot \sin ^{-1} x}{\sqrt{1-x^2}}\right)-\sin ^{-1} x \cdot\left(\frac{1 \cdot \sqrt{1-x^2}+\frac{x \cdot 2 x}{2 \sqrt{1-x^2}}}{1-x^2}\right) \\ & =\frac{x}{\sqrt{1-x^2}} \cdot \frac{1}{\sqrt{1-x^2}} \\ & =\frac{\sin ^{-1}…
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