Question

If $\int f\left(x\right)dx=-2\cos \sqrt{x}+c$, then $f\left(x\right)$ is equal to

• A. $\sin \sqrt{x}$
• B. $\frac{\sin \sqrt{x}}{\sqrt{x}}$
• C. $2\cos \sqrt{x}$
• D. None of these

Answer 1

The correct option is B $\frac{\sin \sqrt{x}}{\sqrt{x}}$

$\int f\left(x\right)dx=-2\cos \sqrt{x}+c$

$\therefore \frac{d}{dx}(-2\cos \sqrt{x}+c)=f\left(x\right)$

$\Rightarrow \frac{d}{dx}(-2\cos \sqrt{x})+\frac{d}{dx}\left(c\right)=f\left(x\right)$

$\Rightarrow -2(-\sin \sqrt{x})\frac{d}{x}\sqrt{x}+0=f\left(x\right)$

$\Rightarrow 2\sin \sqrt{x}\left(\frac{1}{2\sqrt{x}}\right)=f\left(x\right)$

$\therefore f\left(x\right)=\frac{\sin \sqrt{x}}{\sqrt{x}}$