Definite Integration
Odd/Even Function Splitting
nta_pyq_2024_apr
Grade 12
Question:
The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^2 y}\,dy$ is:
$2\pi^2$
$\dfrac{\pi^2}{2}$
$\dfrac{\pi}{2}$
$\pi^2$
Step-by-Step Solution
Key Concept: Split: $\int_{-\pi}^{\pi}\frac{2y}{1+\cos^2y}dy$ (odd function, integral = 0) $+\int_{-\pi}^{\pi}\frac{2y\sin y}{1+\cos^2y}dy$ (even). Use King's on the even part.
$I=\pi^2$.
Correct Answer: 4