Question

The value of $\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx$ is
(a) 2 cos $\sqrt{x}$ + C

(b) $\sqrt{\frac{\cos x}{x}}+C$

(c) sin $\sqrt{x}+C$

(d) 2 sin $\sqrt{x}+C$

Answer 1

(d) 2 sin $\sqrt{x}+C$

$$\begin{gathered} \mathrm{Let}I=\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx \\ \mathrm{Putting}\sqrt{x}=t \\ \Rightarrow \frac{1}{2\sqrt{x}}dx=dt \\ \Rightarrow \frac{dx}{\sqrt{x}}=2dt \\ \therefore I=2\int \cos t·dt \\ =2\sin t+C \\ =2\sin \sqrt{x}+C\left(\because \mathit{}t=\sqrt{x}\right) \end{gathered}$$