Question:
The value of the integral \(\int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx\) is equal to:
The value of the integral \(\int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx\) is equal to:
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For any integral of the form $\int_{0}^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx$ or $\int_{0}^{\pi/2} \frac{\cos^n x}{\sin^n x + \cos^n x} \, dx$, the answer is always $\frac{\pi}{4}$, regardless of the value of $n$.
Updated On: Jun 3, 2026
- $\pi$
- $\pi/2$
- $\pi/4$
- 0
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The Correct Option is C
Solution and Explanation
Concept:
This problem can be solved using the property of definite integrals:
$$\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$$
Step 1: Define the integral and apply the property.
Let $I = \int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx \quad \text{---(1)}$ Applying the property $x \rightarrow \frac{\pi}{2} - x$: $$I = \int_{0}^{\pi/2} \frac{\sin^{3/2} (\frac{\pi}{2} - x)}{\sin^{3/2} (\frac{\pi}{2} - x) + \cos^{3/2} (\frac{\pi}{2} - x)} \, dx$$ Since $\sin(\frac{\pi}{2} - x) = \cos x$ and $\cos(\frac{\pi}{2} - x) = \sin x$, the integral becomes: $$I = \int_{0}^{\pi/2} \frac{\cos^{3/2} x}{\cos^{3/2} x + \sin^{3/2} x} \, dx \quad \text{---(2)}$$
Step 2: Combine the two expressions.
Adding equations (1) and (2): $$2I = \int_{0}^{\pi/2} \left( \frac{\sin^{3/2} x + \cos^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \right) dx$$ $$2I = \int_{0}^{\pi/2} 1 \, dx$$
Step 3: Final integration.
$$2I = [x]_{0}^{\pi/2}$$ $$2I = \frac{\pi}{2} - 0$$ $$I = \frac{\pi}{4}$$
Step 1: Define the integral and apply the property.
Let $I = \int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx \quad \text{---(1)}$ Applying the property $x \rightarrow \frac{\pi}{2} - x$: $$I = \int_{0}^{\pi/2} \frac{\sin^{3/2} (\frac{\pi}{2} - x)}{\sin^{3/2} (\frac{\pi}{2} - x) + \cos^{3/2} (\frac{\pi}{2} - x)} \, dx$$ Since $\sin(\frac{\pi}{2} - x) = \cos x$ and $\cos(\frac{\pi}{2} - x) = \sin x$, the integral becomes: $$I = \int_{0}^{\pi/2} \frac{\cos^{3/2} x}{\cos^{3/2} x + \sin^{3/2} x} \, dx \quad \text{---(2)}$$
Step 2: Combine the two expressions.
Adding equations (1) and (2): $$2I = \int_{0}^{\pi/2} \left( \frac{\sin^{3/2} x + \cos^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \right) dx$$ $$2I = \int_{0}^{\pi/2} 1 \, dx$$
Step 3: Final integration.
$$2I = [x]_{0}^{\pi/2}$$ $$2I = \frac{\pi}{2} - 0$$ $$I = \frac{\pi}{4}$$
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