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