Question

$\displaystyle\int_{0}^{\pi/2} \dfrac{\sin^{n}\theta}{\sin^{n}\theta + \cos^{n}\theta}\,d\theta$ is equal to

• A. 1
• B. 0
• C. $\dfrac{\pi}{2}$
• D. $\dfrac{\pi}{4}$

Hint:

Standard result: $\displaystyle\int_0^{\pi/2}\frac{\sin^n x}{\sin^n x + \cos^n x}\,dx = \frac{\pi}{4}$ for any $n>0$. This follows from the King's property of definite integrals.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Use the symmetry property $\displaystyle\int_0^{\pi/2} f(\sin\theta)\,d\theta = \int_0^{\pi/2} f(\cos\theta)\,d\theta$.
Step 2: Detailed Explanation:
Let $I = \displaystyle\int_0^{\pi/2}\frac{\sin^n\theta}{\sin^n\theta+\cos^n\theta}\,d\theta$. By the property (substituting $\theta \to \frac{\pi}{2}-\theta$):
$I = \displaystyle\int_0^{\pi/2}\frac{\cos^n\theta}{\cos^n\theta+\sin^n\theta}\,d\theta$.
Adding: $2I = \displaystyle\int_0^{\pi/2}1\,d\theta = \frac{\pi}{2}$, so $I = \dfrac{\pi}{4}$.
Step 3: Final Answer:
The integral equals $\dfrac{\pi}{4}$.