Question

Evaluate the integral \( \displaystyle \int_{0}^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx \).

• A. \(0\)
• B. \( \dfrac{\pi}{4} \)
• C. \( \dfrac{\pi}{2} \)
• D. \( \pi \)

Hint:

For many definite integrals between \(0\) and \( \frac{\pi}{2} \), replacing \(x\) with \( \frac{\pi}{2}-x \) helps exploit symmetry between \( \sin x \) and \( \cos x \).

Answer 1

The Correct Option is B

Concept: For definite integrals of the form \[ \int_0^{a} f(x)\,dx \] we can use the property: \[ \int_0^{a} f(x)\,dx = \int_0^{a} f(a-x)\,dx \] This symmetry property is very useful for expressions involving \( \sin x \) and \( \cos x \).
Step 1: {Let} \[ I = \int_{0}^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx \]
Step 2: {Replace \(x\) with \( \frac{\pi}{2}-x \).} \[ I = \int_{0}^{\pi/2} \frac{\cos^n x}{\cos^n x + \sin^n x} \, dx \]
Step 3: {Add the two integrals.} \[ 2I = \int_{0}^{\pi/2} \left( \frac{\sin^n x}{\sin^n x+\cos^n x} + \frac{\cos^n x}{\sin^n x+\cos^n x} \right) dx \] \[ 2I = \int_{0}^{\pi/2} \frac{\sin^n x + \cos^n x}{\sin^n x+\cos^n x} \, dx \] \[ 2I = \int_{0}^{\pi/2} 1\,dx \]
Step 4: {Evaluate the integral.} \[ 2I = \frac{\pi}{2} \] \[ I = \frac{\pi}{4} \]