CUET · MATHS · PYQ PAPER 2023
The value of the integral \(\int_0^\pi 2 x \sin ^3 x\ d x\) is:
- A \(\frac{2 \pi}{3}\)
- B \(\pi\)
- C \(\frac{4 \pi}{3}\)
- D \(\frac{5 \pi}{3}\)
Answer & Solution
Correct Answer
(C) \(\frac{4 \pi}{3}\)
Step-by-step Solution
Detailed explanation
\(I = \int_0^\pi 2 x \sin ^3 x\ d x\) \(I = \int_0^\pi 2 (\pi - x) \sin ^3 (\pi - x)\ d x = \int_0^\pi (2\pi - 2x) \sin ^3 x\ d x\) \(2I = \int_0^\pi 2\pi \sin ^3 x\ d x \implies I = \pi \int_0^\pi \sin ^3 x\ d x\)…
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