Definite Integration
Odd Function Splitting — Symmetric Integral
nta_pyq_2026_jan
Grade 12
Question:
The value of $\displaystyle\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin\!(|x|+\pi/6)}\right)dx$ is equal to:
Step-by-Step Solution
Key Concept: Split: $\pi\int_{-\pi/6}^{\pi/6}\frac{dx}{1-\sin(|x|+\pi/6)}+4\int_{-\pi/6}^{\pi/6}\frac{x^{11}}{1-\sin(|x|+\pi/6)}\,dx$. The second integral is 0 (odd function times even denominator on symmetric interval).
$I=4\pi$.
Correct Answer: 4