Integral of $$$\frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}}$$$

Find $$$\int \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\, dx$$$.

Let $$$u=\sqrt{x}$$$.

Then $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (steps can be seen »), and we have that $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

The integral can be rewritten as

$${\color{red}{\int{\frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}} d x}}} = {\color{red}{\int{2 \cos{\left(u \right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=2$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{2 \cos{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\cos{\left(u \right)} d u}\right)}}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$2 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 2 {\color{red}{\sin{\left(u \right)}}}$$

Recall that $$$u=\sqrt{x}$$$:

$$2 \sin{\left({\color{red}{u}} \right)} = 2 \sin{\left({\color{red}{\sqrt{x}}} \right)}$$

Therefore,

$$\int{\frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}} d x} = 2 \sin{\left(\sqrt{x} \right)}$$

Add the constant of integration:

$$\int{\frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}} d x} = 2 \sin{\left(\sqrt{x} \right)}+C$$

$$$\int \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\, dx = 2 \sin{\left(\sqrt{x} \right)} + C$$$A