Let f n =∫₀^((π)/2)(Σ k=1 ⁿ sin ^(k-1) x)(Σ k=1 ⁿ(2 k-1) sin ^(k-1) x) cos x d x, n N. Then f₂₁-f₂₀ is equal to _ _ _ _ _

Integral Calculus: JEE Main 2023 Mathematics question

JEE Main 2023 · 13 Apr, Evening shift · Q33

JEE Main 202313 Apr, Evening shiftMathematicsMediumIntegral Calculus369× since 2002

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 \mathbb{N}

. Then

f_{21}-f_{20}

is equal to _________

Your answerType your answer. Integers and decimals are accepted.

JEE Main 2023 Integral Calculus Question with Answer (13 Apr Evening shift, Q33) | JEE Arc

JEE Main 2023 · 13 Apr, Evening shift · Q33

JEE Main 202313 Apr, Evening shiftMathematicsMediumIntegral Calculus369× since 2002

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 \mathbb{N}

. Then

f_{21}-f_{20}

is equal to _________

Your answerType your answer. Integers and decimals are accepted.