Definite Integration
Even/Odd Split of Integral
nta_pyq_2023_jan
Grade None
Question:
The value of the integral $\displaystyle\int_{-\pi/4}^{\pi/4}\dfrac{x+\pi/4}{2-\cos2x}\,dx$ is:
\dfrac{\pi^2}{6}
\dfrac{\pi^2}{12\sqrt{3}}
\dfrac{\pi^2}{3\sqrt{3}}
\dfrac{\pi^2}{6\sqrt{3}}
Step-by-Step Solution
Key Concept: Split: $\int_{-\pi/4}^{\pi/4}\frac{x}{2-\cos2x}dx+\frac{\pi}{4}\int_{-\pi/4}^{\pi/4}\frac{dx}{2-\cos2x}$. First is 0 (odd/even). $2-\cos2x=1+2\sin^2x$.
$\dfrac{\pi^2}{6\sqrt{3}}$.
Correct Answer: 4