Find Integral of e^3x | e^3x Integration

The integration of e^3x is e^3x/3. In this post, we will learn how to find the integral of e to the 3x. Let us recall the formula of the integral of e^mx:

$\int e^{mx} dx=\dfrac{e^{mx}}{m}+C$ where C is an integral constant. Thus, the integral of e^3x will be equal to $\int e^{3x} dx=\dfrac{e^{3x}}{3}+C$.

What is the Integration of e^3x

Question: What is the integration of e^3x?

Answer: The integration of e^3x is $\dfrac{e^{3x}}{3}$.

Explanation:

Step 1: We will put $z=3x$.

Differentiating both sides of $z=3x$, we have

$\dfrac{dz}{dx}=3$

$\Rightarrow dx= \dfrac{dz}{3}$

Step 2: $\therefore \int e^{3x} dx = \int e^z \dfrac{dz}{3}$

$=\dfrac{1}{3} \int e^z dz$

$=\dfrac{1}{3} e^z +C$ as the integration of e^x is e^x

$=\dfrac{1}{3} e^{3x}+C$ as $z=3x$.

Conclusion: Thus, the integration of e^3xdx is $\dfrac{1}{3} e^{3x}+C$.

Definite integral of e^3x

Question: Find the definite integral $\int_0^1 e^{3x} dx$.

Answer:

We have shown above that the integration of $e^{3x} dx$ is $\dfrac{1}{3} e^{3x}$. Thus, we have that

$\int_0^1 e^{3x} dx$

$=[\dfrac{1}{3} e^{3x}]_0^1$

$=\dfrac{1}{3}[e^{3x}]_0^1$

$=\dfrac{1}{3}(e^{3 \cdot 1} -e^0)$

$=\dfrac{1}{3}(e^3 -1)$

So the definite integration of e^3x from 0 to 1 is equal to (e³-1)/3.

Also Read: 

Derivative of 2^x

Derivative of $\pi$

Derivative of sin4x

Derivative of e^1/x

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