JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f_n=\int \limits_0^{\frac{\pi}{2}}\left(\sum \limits_{k=1}^n \sin ^{k-1} x\right)\left(\sum \limits_{k=1}^n(2 k-1) \sin ^{k-1} x\right) \cos x\) \(d x, n \in N\). Then \(f_{21}-f_{20}\) is equal to \(...........\).
- A \(40\)
- B \(41\)
- C \(42\)
- D \(43\)
Answer & Solution
Correct Answer
(B) \(41\)
Step-by-step Solution
Detailed explanation
\(f _{ n }( x )=\int \limits_0^{\frac{\pi}{2}}\left(1+\sin x+\sin ^2 x+\sin ^3 x+\ldots+\sin ^{n-1}(x)\right)\) \(\left(1+3 \sin x +5 \sin ^2 x +\ldots+(2 n -1)\right) \sin ^{ n -1} x \cdot \cos x d x\) Multiply and divide by \(\sqrt{\sin x}\)…
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