Question

The value of the integral $\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x+\frac{\pi}{4}}{2-\cos 2 x} d x$ is :

• A. $\frac{\pi^2}{12 \sqrt{3}}$
• B. $\frac{\pi^2}{6}$
• C. $\frac{\pi^2}{3 \sqrt{3}}$
• D. $\frac{\pi^2}{6 \sqrt{3}}$

Answer 1

Answer: (D)

$\frac{\pi^2}{6 \sqrt{3}}$

Answer 2

I=4−π​∫4π​​2−cos2xx+4π​​dx
x→−x
I=−4π​∫4π​​2−cos2x−x+4π​​dx
(1)+(2)
2I=4−π​∫4π​​2−cos2x2π​​dx
I=4π​⋅20∫4π​​2−cos2xdx​dx
I=4π​⋅20∫4π​​2(1+tan2x)−(1−tan2x)(1+tan2x)dx​
I=4π​0∫1​3t2+1dt​
⇒I=23​π​tan−13​
I=63​π2​