JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of the integral \(\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} {{{\sin }^4}\,x\left( {1 + \log \left( {\frac{{2 + \sin \,x}}{{2 - \sin \,x}}} \right)} \right)\,dx} \) is
- A \(\frac{3}{{16}}\pi \)
- B \(0\)
- C \(\frac{3}{{8}}\pi \)
- D \(\frac{3}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{3}{{8}}\pi \)
Step-by-step Solution
Detailed explanation
\(I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{4} x\left(1+\log \left(\frac{2+\sin x}{2-\sin x}\right)\right) d x\) \(\left.\Rightarrow I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{4}(-x)\right)\left(1+\log \left(\frac{2+\sin (-x)}{2-\sin (-x)}\right)\right) \cdot d x\)…
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