AP EAMCET · Maths · Definite Integration
If \(I_n=\int_0^{\pi / 2} \sin ^n(x) d x\) and \(I_n=(k) I_{n-2}\), then what will be the value of \(k\) ?
- A \(\frac{n}{n-1}\)
- B \(\frac{n-1}{n}\)
- C \(\frac{n+1}{n}\)
- D \(\frac{n}{n+1}\)
Answer & Solution
Correct Answer
(B) \(\frac{n-1}{n}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} \mathrm{I}_n & =\int_0^{\pi / 2} \sin x \cdot \sin ^{n-1} x \mathrm{dx} \\ & \left.=\sin ^{n-1} x(-\cos x)\right]_0^{\pi / 2}+\int_0^{\pi / 2}(n-1) \cdot \sin ^{n-2} x \\ & =0+(n-1) \int_0^{\pi / 2} \sin ^{n-2} x\left(1-\sin ^2 x\right) d x \\ \mathrm{I}_n &…
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