AP EAMCET · Maths · Definite Integration
\(\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x=\)
- A \(2 \pi(1-\log 3)\)
- B \(2 \pi\left(1-\frac{3}{4} \log 3\right)\)
- C \(\pi\left(1-\frac{3}{4} \log 3\right)\)
- D \(4 \pi(1-\log 3)\)
Answer & Solution
Correct Answer
(B) \(2 \pi\left(1-\frac{3}{4} \log 3\right)\)
Step-by-step Solution
Detailed explanation
\(I=\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x \Rightarrow I=2 \int_0^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x\) ....(i) \(I=2 \int_0^\pi \frac{(\pi-x) \sin ^3(\pi-x)}{4-\cos ^2(\pi-x)} d x \Rightarrow I=2 \int_0^\pi \frac{(\pi-x) \sin ^3 x}{4-\cos ^2 x} d x\) ....(ii)…
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