AP EAMCET · Maths · Indefinite Integration
\(\int \frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sqrt{x}\left(\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}\right)} d x=\)
- A \(\frac{2}{\pi}\left[\sin ^{-1} \sqrt{x}(2 x-1)+\sqrt{x(1-x)}\right]+x+C\)
- B \(\frac{8}{\pi}\left(\left(\sqrt{x} \sin ^{-1} \sqrt{x}-\sqrt{1-x}\right)\right)-2 \sqrt{x}+C\)
- C \(\frac{2}{\pi}\left[(2 x-1) \sin ^{-1} \sqrt{x}-\sqrt{x(1-x)}\right]-x+C\)
- D \(\frac{2}{\pi}\left[(2 x-1) \sin ^{-1} \sqrt{x}-\sqrt{x(1-x)}\right]+x+C\)
Answer & Solution
Correct Answer
(B) \(\frac{8}{\pi}\left(\left(\sqrt{x} \sin ^{-1} \sqrt{x}-\sqrt{1-x}\right)\right)-2 \sqrt{x}+C\)
Step-by-step Solution
Detailed explanation
Let \(I=\int \frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sqrt{x}\left(\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}\right)} d x\)…
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