Question

$\underset{0}{\overset{{\mathrm{\pi }}^2/4}{\int }}\frac{\sin \sqrt{x}}{\sqrt{x}}dx$ equals
(a) 2
(b) 1
(c) π/4
(d) π²/8

Answer 1

(a) 2

$$\begin{gathered} {\int }_0^{\frac{{\mathrm{\pi }}^2}{4}}\frac{\sin \sqrt{x}}{\sqrt{x}}dx \\ \mathrm{Let}\sqrt{x}=t,\mathrm{then}\frac{1}{2\sqrt{x}}dx=dt \\ \mathrm{When}x=0,t=0,x=\frac{{\mathrm{\pi }}^2}{4},t=\frac{\mathrm{\pi }}{2} \\ \mathrm{Therefore}\mathrm{the}\mathrm{integral}\mathrm{becomes} \\ {\int }_0^{\frac{\mathrm{\pi }}{2}}2\mathrm{sint}\mathrm{dt} \\ =-2{\left[\mathrm{cost}\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ =2 \end{gathered}$$