\[\begin{align*} \boxed{\int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) + C} \end{align*}\]

where \( C \) is the constant of integration.

Step-by-Step Solution:

In this article, we will go through the steps to find the integral of \( \cos(2x) \). We will proceed with $u$-substitution as we know the common integral $\cos x$.

\[\begin{align*} \int \cos(2x) \, dx \end{align*}\]

Since \( 2x \) is inside the cosine function, we’ll set our $u = 2x$

\[\begin{align*} \dfrac{du}{dx} = 2 \Rightarrow dx = \dfrac{du}{2} \end{align*}\]

Now, substituting \( u \) and \( dx \):

\[\begin{align*} \int \cos(2x) \, dx = \int \cos(u) \cdot \dfrac{du}{2} \end{align*}\]

This simplifies to:

\[\begin{align*} \dfrac{1}{2} \int \cos(u) \, du \end{align*}\]

The integral of \( \cos(u) \) is \( \sin(u) \), so we have:

\[\begin{align*} \dfrac{1}{2} \sin(u) + C \end{align*}\]

Finally, replace \( u \) with \( 2x \) in the answer:

\[\begin{align*} \dfrac{1}{2} \sin(2x) + C \end{align*}\]

So, the integral of \( \cos(2x) \) is:

\[\begin{align*} \boxed{\int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) + C} \end{align*}\]

Why This Works:

The reason this works is that substitution helps us simplify the integral by converting \( \cos(2x) \) into a more recognizable integral, \( \cos(u) \). After integrating \( \cos(u) \) as \( \sin(u) \), we reintroduce the original variable \( x \). The result, \( \dfrac{1}{2} \sin(2x) \), accounts for the coefficient in the argument of the cosine function.

Final Answer:

To conclude, the integral of \( \cos(2x) \) is:

\[\begin{align*} \boxed{\int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) + C} \end{align*}\]

$u$-substitution is a key technique when solving integrals as it uses your memory of other integrals to do the heavy lifting!