Question

The value of $\underset{3}{\overset{8}{\int }}\frac{\sin \sqrt{x+1}}{\sqrt{x+1}}dx$ is

• A. $2\cos 2-2\cos 3$
• B. $1$
• C. $\cos 2-\cos 3$
• D. $2\cos 2+2\cos 3$

Answer 1

The correct option is A $2\cos 2-2\cos 3$

$I=\underset{3}{\overset{8}{\int }}\frac{\sin \sqrt{x+1}}{\sqrt{x+1}}dx$
Put $\sqrt{x+1}=t\Rightarrow \frac{1}{\sqrt{x+1}}dx=2dt$
$\Rightarrow I=\underset{2}{\overset{3}{\int }}2\sin tdt=-2\cos t{\mid }_2^3$
$=-2(\cos 3-\cos 2)$
$=2\cos 2-2\cos 3$