The given integral is:

$\int e^{5x} \, dx$

This integral can be solved directly without the need for integration by parts. However, if you want to express it in the context of integration by parts, let's break it down.

Applying Integration by Parts:

For integration by parts, we use the formula:

$\int u \, dv = uv - \int v \, du$

However, in this particular case, it's easier to recognize that $e^{5x}$ is best integrated directly.

Choosing $u$ and $dv$:

We could choose $u = 1$ and $dv = e^{5x} \, dx$.

Then:

• $du = 0 \, dx$
• $v = \frac{e^{5x}}{5}$ (because the integral of $e^{5x}$ with respect to $x$ is $\frac{e^{5x}}{5}$)

Integration:

Using integration by parts here is somewhat unnecessary because:

$\int e^{5x} \, dx = \frac{e^{5x}}{5} + C$

Components for Integration by Parts (if needed):

• $u = 1$
• $dv = e^{5x} \, dx$
• $du = 0 \, dx$
• $v = \frac{e^{5x}}{5}$

But again, in this case, directly integrating $e^{5x}$ is the simplest approach.

Would you like to explore integration by parts further, or do you have any questions? Here are some related questions for further exploration:

1. How do you integrate functions of the form $e^{ax}$ where $a$ is a constant?
2. Why is it not necessary to use integration by parts for simple exponential functions?
3. Can you explain the process of finding the antiderivative of $e^{5x}$?
4. What are some examples of integrals where integration by parts is necessary?
5. How would you apply integration by parts to the integral $\int x e^{5x} \, dx$?

Tip: Always consider the simplest integration technique first before applying more complex methods like integration by parts.