Integral of $$$7 \cos{\left(7 x \right)}$$$

Find $$$\int 7 \cos{\left(7 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=7$$$ and $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$:

$${\color{red}{\int{7 \cos{\left(7 x \right)} d x}}} = {\color{red}{\left(7 \int{\cos{\left(7 x \right)} d x}\right)}}$$

Let $$$u=7 x$$$.

Then $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{7}$$$.

So,

$$7 {\color{red}{\int{\cos{\left(7 x \right)} d x}}} = 7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{7}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}} = 7 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}$$

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

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

Recall that $$$u=7 x$$$:

$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{\left(7 x\right)}} \right)}$$

Therefore,

$$\int{7 \cos{\left(7 x \right)} d x} = \sin{\left(7 x \right)}$$

Add the constant of integration:

$$\int{7 \cos{\left(7 x \right)} d x} = \sin{\left(7 x \right)}+C$$

$$$\int 7 \cos{\left(7 x \right)}\, dx = \sin{\left(7 x \right)} + C$$$A