AP EAMCET · Maths · Definite Integration
\(\int_0^\pi x \sin ^3 x \cos ^2 x d x=\)
- A \(\frac{2 \pi}{15}\)
- B \(\frac{4 \pi}{15}\)
- C \(\frac{\pi}{30}\)
- D \(\frac{2 \pi}{5}\)
Answer & Solution
Correct Answer
(A) \(\frac{2 \pi}{15}\)
Step-by-step Solution
Detailed explanation
Let \(\int_0^\pi x \sin ^3 x \cos ^2 x d x=I\) ...(i) \(\Rightarrow I=\int_0^\pi(\pi-x) \sin ^3(\pi-x) \cos ^2(\pi-x) d x\) \(I=\int_0^\pi(\pi-x) \sin ^3(x) \cos ^2 x d x\) ...(ii) Equation (i) + (ii) \(2 I=\pi \int_0^\pi \sin ^3 x \cos ^2 x d x\)…
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