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

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

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 $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Thus,

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

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

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

For the integral $$$\int{u \cos{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Let $$$\operatorname{\omega}=u$$$ and $$$\operatorname{dv}=\cos{\left(u \right)} du$$$.

Then $$$\operatorname{d\omega}=\left(u\right)^{\prime }du=1 du$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\cos{\left(u \right)} d u}=\sin{\left(u \right)}$$$ (steps can be seen »).

Thus,

$$2 {\color{red}{\int{u \cos{\left(u \right)} d u}}}=2 {\color{red}{\left(u \cdot \sin{\left(u \right)}-\int{\sin{\left(u \right)} \cdot 1 d u}\right)}}=2 {\color{red}{\left(u \sin{\left(u \right)} - \int{\sin{\left(u \right)} d u}\right)}}$$

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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