MHT CET · Maths · Definite Integration
The value of \(\int_{0}^{\pi} x \sin ^{3} x d x\) is
- A \(\frac{4 \pi}{3}\)
- B \(\frac{2 \pi}{3}\)
- C 0
- D None of these
Answer & Solution
Correct Answer
(B) \(\frac{2 \pi}{3}\)
Step-by-step Solution
Detailed explanation
Let \(I=\int_{0}^{\pi} x \sin ^{3} x d x\) ...(i)
Also, \(I=\int_{0}^{\pi}(\pi-x) \sin ^{3} x d x\) ...(ii)
On adding Eqs. (i) and (ii), we get
\(2 I =\pi \int_{0}^{\pi} \sin ^{3} x d x \)
\( =\frac{\pi}{4} \int_{0}^{\pi}(3 \sin x-\sin 3 x) d x \)
\( =\frac{\pi}{4}\left[-3 \cos x+\frac{\cos 3 x}{3}\right]_{0}^{\pi} \)
\( =\frac{\pi}{4}\left[3-\frac{1}{3}+3-\frac{1}{3}\right]=\frac{4 \pi}{3} \)
\( \text { Hence, } I =\frac{2 \pi}{3} \)
Also, \(I=\int_{0}^{\pi}(\pi-x) \sin ^{3} x d x\) ...(ii)
On adding Eqs. (i) and (ii), we get
\(2 I =\pi \int_{0}^{\pi} \sin ^{3} x d x \)
\( =\frac{\pi}{4} \int_{0}^{\pi}(3 \sin x-\sin 3 x) d x \)
\( =\frac{\pi}{4}\left[-3 \cos x+\frac{\cos 3 x}{3}\right]_{0}^{\pi} \)
\( =\frac{\pi}{4}\left[3-\frac{1}{3}+3-\frac{1}{3}\right]=\frac{4 \pi}{3} \)
\( \text { Hence, } I =\frac{2 \pi}{3} \)
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