The integration of e2x is equal to e2x/2. In this post, we will learn how to integrate e to the 2x. The integration of e2x can be derived directly from the formula of the integral of emx which is given below:
$\int e^{mx} dx=\dfrac{e^{mx}}{m}+C$ where C is an integration constant.
Putting m=2 in the above formula, the integral of e to the 2x is equal to
$\int e^{2x} dx=\dfrac{e^{2x}}{2}+C$.

What is the Integration of e2x
Question: What is the integration of e2x?
Answer: The integration of e2x is $\dfrac{e^{2x}}{2}$.
Explanation:
We will find the integration of e2x by the substitution method, we will follow the below steps.
Step 1: Let us put $t=2x$.
Differentiating both sides with respect to x, we have
$\dfrac{dt}{dx}=2$
$\Rightarrow dx= \dfrac{dt}{2}$
Step 2: Let us now put the values of 2x and dx in the original integration. By doing so, we will get that
$\therefore \int e^{2x} dx = \int e^t \dfrac{dt}{2}$
$=\dfrac{1}{2} \int e^t dt$ as 1/2 is a constant, so we can take it out of the integration.
$=\dfrac{1}{2} e^t +C$ as the integration of ex is ex
$=\dfrac{1}{2} e^{2x}+C$ as t=2x.
So the integration of e2xdx is $\dfrac{1}{2} e^{2x}+C$ where C is an integral constant.
More Integrals: Integral of e3x
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Definite Integral of e2x
Question: Find the definite integral $\int_0^1 e^{2x} dx$.
Answer:
We have shown above that the integration of $e^{2x} dx$ is $\dfrac{1}{2} e^{2x}$. Thus, we have that
$\int_0^1 e^{2x} dx$
$=[\dfrac{1}{2} e^{2x}]_0^1$
$=\dfrac{1}{2}[e^{2x}]_0^1$
$=\dfrac{1}{2}(e^{2 \cdot 1} -e^0)$
$=\dfrac{1}{2}(e^2 -1)$
So the definite integration of e2x from 0 to 1 is equal to (e2-1)/2.
Also Read:
Derivative of $\sqrt{x}+\dfrac{1}{\sqrt{x}}$
FAQs
Q1: What is the integration of e2x?
Answer: The integration of e2x is e2x/2 +C where C is an integral constant.