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Solve the following.
Evaluate: ∫₀^(π/2) (sin^(3/2) x)/(sin^(3/2) x + cos^(3/2) x) dx
Evaluate: $\int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$
Answer
Answer
Integral: $\int_{0}^{\frac{\pi}{2}} \left(\frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x}\right) dx$
Let $I = \int_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x}dx$.
Using $\int_0^a f(x)dx = \int_0^a f(a-x)dx$: $I = \int_0^{\pi/2} \frac{\cos^{3/2} x}{\cos^{3/2} x + \sin^{3/2} x}dx$.
Adding: $2I = \int_0^{\pi/2} 1\,dx = \frac{\pi}{2}$.
Hence $I = \frac{\pi}{4}$.
Final answer: $\frac{\pi}{4}$