Question

Evaluate:$$\int \frac{\sin \sqrt{x}}{\sqrt{x}}\text{ dx }$$

Answer 1

$$\text{ Let I }=\int \frac{\sin \sqrt{x}}{\sqrt{x}}dx$$
$$\text{ Let }\sqrt{x}=t$$
$$\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$$
$$\Rightarrow \frac{dx}{\sqrt{x}}=2dt$$
$$\text{ Putting }\sqrt{x}=t\text{ and }\frac{dx}{\sqrt{x}}=2\text{ dt , we get}$$
$$\therefore I=2\int \text{ sin t dt}$$
$$=-2\text{ cos t}+C(\because t=\sqrt{x})$$
$$=-2\cos \sqrt{x}+C$$