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

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:

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