Question

$\int_{0}^{\pi/2} \frac{\sin^{100}x}{\sin^{100}x + \cos^{100}x} dx =$

• A. $\pi/2$
• B. $\pi/4$
• C. 100
• D. 50

Hint:

For symmetric integrals of the form $\frac{f(\sin)}{f(\sin)+f(\cos)}$ from $0$ to $\pi/2$, the answer is always $(Upper Limit - Lower Limit) / 2$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.

Step 2: Meaning
Let $I$ be the given integral. Replacing $x$ with $\pi/2 - x$ gives a second form for $I$.

Step 3: Analysis
$I = \int_{0}^{\pi/2} \frac{\cos^{100}x}{\cos^{100}x + \sin^{100}x} dx$. Adding the two forms of $I$: $2I = \int_{0}^{\pi/2} \frac{\sin^{100}x + \cos^{100}x}{\sin^{100}x + \cos^{100}x} dx = \int_{0}^{\pi/2} 1 dx = \pi/2$.

Step 4: Conclusion
$2I = \pi/2 \implies I = \pi/4$.
Final Answer: (B)