Question

$\int \frac{\sqrt{x}\cos x}{\sqrt{x}}dx$

• A. $2\sin \exp \sqrt{x}+c$
• B. $2\cos \exp \sqrt{x}+c$
• C. $\sin \exp \sqrt{x}+c$
• D. $\sin e^x+c$

Answer 1

The correct option is A $2\sin \exp \sqrt{x}+c$

Substitute $\sqrt{x}=t$ $\frac{1}{2}\frac{1}{\sqrt{x}}.dx=dt$
$\Rightarrow \frac{1}{\sqrt{x}}dx=2dt$
$\int 2e^{t}cos{e}^{t}dt.$
Substitute $e^t=p$
$\Rightarrow e^t\cdot dt=dp.$
$2\int \cos pdp=2\sin p+c$
$\sin e^t+c$
$=sin{e}^{\sqrt{x}}+c$
So, $\int \frac{e^{\sqrt{x}}cos{e}^{\sqrt{x}}}{\sqrt{x}}dx=2sin{e}^{\sqrt{x}}+c.$