Definite Integration
Sum of Products of Geometric Series Integrals
nta_pyq_2023_apr
Grade 12
Question:
Let $f_n=\displaystyle\int_0^{\pi/2}\!\left(\sum_{k=1}^n\sin^{k-1}x\right)\!\left(\sum_{k=1}^n(2k-1)\sin^{k-1}x\right)\cos x\,dx$, $n\in\mathbb{N}$. Then $f_{21}-f_{20}$ is equal to
Step-by-Step Solution
Key Concept: Substitute $\sin x=t$. The integral becomes $\int_0^1(1+t+\cdots+t^{n-1})(1+3t+\cdots+(2n-1)t^{n-1})dt$. Show $f_n=n^2$ via a substitution $z=t^{1/2}+\cdots$.
$f_n=n^2$. $f_{21}-f_{20}=41$.
Correct Answer: 41