Question

Evaluate the definite integral: \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x+\frac{\pi}{2}} {2-\sin^2 x} \,dx \]

• A. \[ \frac{\pi^2}{6\sqrt3} \]
• B. \[ \frac{\pi}{\sqrt2} \tan^{-1} \left( \frac{\sqrt3}{\sqrt2} \right) \]
• C. \[ \frac{\pi^2}{3\sqrt2} \]
• D. \[ \frac{\pi}{2} \tan^{-1} \left( \frac{\sqrt3}{2} \right) \]

Hint:

Whenever a definite integral has symmetric limits \([-a,a]\), first test whether the integrand is odd or even. This often eliminates a large portion of the calculation and converts the problem into a much simpler integral.

Answer 1

The Correct Option is B