Question

Calculate the integral: 
\[ \int_{0}^{\pi/4} \sin\sqrt{x}\ dx = \underline{\hspace{1cm}} . \]

Hint:

For integrals with \(\sin(\sqrt{x})\), use the substitution \(u=\sqrt{x}\).

Answer 1

Correct Answer: 2

Substitute: 
\[ u = \sqrt{x},\ x = u^{2},\ dx = 2u\,du \] Limits: 
\[ x = 0 \Rightarrow u = 0, x=\pi/4 $\Rightarrow$ u=\sqrt{\pi}/2 \] Integral becomes: 
\[ \int_{0}^{\sqrt{\pi}/2} 2u\sin u\ du \] Use integration by parts: 
\[ \int u\sin u\ du = -u\cos u + \sin u \] Thus: 
\[ I = 2\left[-u\cos u + \sin u\right]_{0}^{\sqrt{\pi}/2} \] Evaluate numerically ≈ 2.00.