\[\begin{align*} \boxed{\int e^{2x} \, dx = \dfrac{1}{2} e^{2x} + C} \end{align*}\]

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

Step-by-Step Solution:

In this article, we’re going to explore how to find the integral of \( e^{2x} \). The function \( e^{2x} \) involves the exponential base \( e \) raised to the power of \( 2x \). Integrating this type of function involves using $u$-substitution to simplify the exponent.

\[\begin{align*} \int e^{2x} \, dx \end{align*}\]

Since \( 2x \) is in the exponent, we’ll use a substitution to simplify it as we know the common integral $e^x$.

Let:

\[\begin{align*} u = 2x \end{align*}\]

Differentiating \( u \) with respect to \( x \):

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

Rewriting the integral in terms of \( u \):

\[\begin{align*} \int e^{2x} \, dx = \int e^{u} \cdot \dfrac{du}{2} \end{align*}\]

This simplifies to:

\[\begin{align*} \dfrac{1}{2} \int e^{u} \, du \end{align*}\]

The integral of \( e^{u} \) is simply \( e^{u} \), so we have:

\[\begin{align*} \dfrac{1}{2} e^{u} + C \end{align*}\]

Finally, substitute \( u = 2x \) back into the expression:

\[\begin{align*} \dfrac{1}{2} e^{2x} + C \end{align*}\]

So, the integral of \( e^{2x} \) is:

\[\begin{align*} \boxed{\int e^{2x} \, dx = \dfrac{1}{2} e^{2x} + C} \end{align*}\]

Why This Works:

When integrating functions like \( e^{2x} \), we often encounter an exponent involving a coefficient (here, the coefficient is 2). Using substitution helps simplify the problem by transforming it into a basic exponential integral.

Final Answer:

To summarize, the integral of \( e^{2x} \) is:

\[\begin{align*} \boxed{\int e^{2x} \, dx = \dfrac{1}{2} e^{2x} + C} \end{align*}\]