Evaluate: ∫₀⁵ (√x)/(√(5-x) + √x) dx (Evaluate)

Evaluate: $\int_{0}^{5} \frac{\sqrt{x}}{\sqrt{5-x} + \sqrt{x}}\,dx$ (Evaluate)

Answer

Official answer key

Integral: $\int_{0}^{5} \left(\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}\right) dx$

Let $I = \int_0^5 \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}\,dx$. Using $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$: $I = \int_0^5 \frac{\sqrt{5-x}}{\sqrt{x}+\sqrt{5-x}}\,dx$. Adding, $2I = \int_0^5 1\,dx = 5$, so $I = \frac{5}{2}$.

Final answer: 5/2