Definite Integration
Even function integral — IBP
MJAT_TS6_P2
Grade 12
Question:
If $L=\displaystyle\lim_{n\to\infty}\int_{-\pi/2}^{\pi/2}\frac{x^4\cos x^2}{x}\cdot\frac{1}{\ln\!\left(\sum_{r=0}^n x^{-r^2\cdot 2!}\right)}\,dx$, then $2L$ equals:
A) $\pi^2-8$
B) $\pi-8$
C) $\pi^2-4$
D) $\dfrac{\pi^2}{2}-2\sqrt{2}$
Step-by-Step Solution
Key Concept: After simplification of the limit, $L=\int_{-\pi/2}^{\pi/2}x^2\cos x\,dx=2\int_0^{\pi/2}x^2\cos x\,dx$ (even integrand). IBP twice: $=2[(x^2\sin x+2x\cos x-2\sin x)]_0^{\pi/2}=2(\pi^2/4-2)$.
$2L=\pi^2-8$. Answer: **A**.
Correct Answer: A