To find the integral of the function [tex]\(e^{3x}\)[/tex], we apply the integral formula for exponential functions.
Here's a step-by-step solution:
1. Identify the integral to be computed:
[tex]\[
\int e^{3x} \, dx
\][/tex]
2. Recall the integral rule for exponential functions:
For a general exponential function [tex]\(e^{ax}\)[/tex], where [tex]\(a\)[/tex] is a constant, the integral is given by:
[tex]\[
\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
3. Apply the rule to our specific problem:
Here, [tex]\(a = 3\)[/tex]. Therefore, by substituting [tex]\(a = 3\)[/tex] into the formula, we get:
[tex]\[
\int e^{3x} \, dx = \frac{1}{3} e^{3x} + C
\][/tex]
4. Write down the final result:
[tex]\[
\int e^{3x} \, dx = \frac{e^{3x}}{3} + C
\][/tex]
So the integral of [tex]\( e^{3x} \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\(\frac{e^{3x}}{3} + C\)[/tex].
