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

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

Let $$$u=x e^{3}$$$.

Then $$$du=\left(x e^{3}\right)^{\prime }dx = e^{3} dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{e^{3}}$$$.

So,

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

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

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

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

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

Recall that $$$u=x e^{3}$$$:

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

Therefore,

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

Add the constant of integration:

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

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