Integrate integral x dx

The problem asks to integrate $\cos(5x)$.

$$\int \cos(5x) \, dx$$

Let $u = 5x$.
Then, differentiate $u$ with respect to $x$:

$$\frac{du}{dx} = 5$$

Rearrange to find $dx$:

$$dx = \frac{1}{5} \, du$$

$$\int \cos(u) \left( \frac{1}{5} \right) \, du$$ $$= \frac{1}{5} \int \cos(u) \, du$$

The integral of $\cos(u)$ is $\sin(u)$.

$$= \frac{1}{5} \sin(u) + C$$

$$= \frac{1}{5} \sin(5x) + C$$

The final answer is $\frac{1}{5} \sin(5x) + C$.