Question

Evaluate: \[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \]

Hint:

For definite integrals, simplify the integrand using substitution or trigonometric identities before integrating. Ensure the limits of integration are updated when substitution is used.

Answer 1

Let \( I = \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \). 

Using substitution, let \( t = \sqrt{x} \). Then, \[ x = t^2, \quad dx = 2t \, dt, \quad \sqrt{x} = t. \] Substitute into the integral: \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin t}{t} \cdot 2t \, dt = 2 \int_0^{\frac{\pi}{2}} \sin t \, dt. \] 

Simplify: \[ I = 2 \left[ -\cos t \right]_0^{\frac{\pi}{2}}. \] Evaluate: \[ I = 2 \left[ -\cos\left(\frac{\pi}{2}\right) + \cos(0) \right] = 2 \left[ 0 + 1 \right] = 2. \] 

Answer: \[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx = 2. \] \bigskip