Question

Integrate:${\int }_3^8\frac{\sin \sqrt{x+1}}{\sqrt{x+1}}dx$

Answer 1

${\int }_3^8\frac{\sin \sqrt{x+1}}{\sqrt{x+1}}dx$

Let $t=\sqrt{x+1}\Rightarrow dt=\frac{1}{2\sqrt{x+1}}dx\Rightarrow 2dt=\frac{dx}{\sqrt{x+1}}$

$=\frac{1}{2}{\int }_3^8\sin tdt$

$=\frac{1}{2}{[-\cos t]}_3^8$

$=\frac{-1}{2}[\cos 8-\cos 3]$

$=\frac{-1}{2}[0.99026-0.9986]$

$=\frac{-1}{2}\times -0.00834$

$=\frac{0.00834}{2}$

$=0.00417$