Integral of $$$3 x \cos{\left(2 x^{2} \right)}$$$

Find $$$\int 3 x \cos{\left(2 x^{2} \right)}\, dx$$$.

Let $$$u=2 x^{2}$$$.

Then $$$du=\left(2 x^{2}\right)^{\prime }dx = 4 x dx$$$ (steps can be seen »), and we have that $$$x dx = \frac{du}{4}$$$.

So,

$${\color{red}{\int{3 x \cos{\left(2 x^{2} \right)} d x}}} = {\color{red}{\int{\frac{3 \cos{\left(u \right)}}{4} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{3}{4}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{3 \cos{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{3 \int{\cos{\left(u \right)} d u}}{4}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{3 {\color{red}{\sin{\left(u \right)}}}}{4}$$

Recall that $$$u=2 x^{2}$$$:

$$\frac{3 \sin{\left({\color{red}{u}} \right)}}{4} = \frac{3 \sin{\left({\color{red}{\left(2 x^{2}\right)}} \right)}}{4}$$

Therefore,

$$\int{3 x \cos{\left(2 x^{2} \right)} d x} = \frac{3 \sin{\left(2 x^{2} \right)}}{4}$$

Add the constant of integration:

$$\int{3 x \cos{\left(2 x^{2} \right)} d x} = \frac{3 \sin{\left(2 x^{2} \right)}}{4}+C$$

$$$\int 3 x \cos{\left(2 x^{2} \right)}\, dx = \frac{3 \sin{\left(2 x^{2} \right)}}{4} + C$$$A