To integrate $e^{5x} \sin(3x)$ by parts, we use the integration by parts formula:

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

We will choose $u$ and $dv$ from the given integral:

$\int e^{5x} \sin(3x) \, dx$

Let's make the following choices:

• Let $u = \sin(3x)$ (since the derivative of $\sin(3x)$ will simplify the problem),
• Let $dv = e^{5x} \, dx$ (since the integral of $e^{5x}$ is simple).

Now, differentiate and integrate:

• $du = 3\cos(3x) \, dx$,
• $v = \frac{1}{5} e^{5x}$.

Using the integration by parts formula:

$\int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \int \frac{1}{5} e^{5x} 3\cos(3x) \, dx$

Simplify the second integral:

$\int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{5} \int e^{5x} \cos(3x) \, dx$

Now we need to apply integration by parts again to the remaining integral:

$\int e^{5x} \cos(3x) \, dx$

Let’s choose:

• $u = \cos(3x)$,
• $dv = e^{5x} \, dx$.

Then:

• $du = -3\sin(3x) \, dx$,
• $v = \frac{1}{5} e^{5x}$.

Now, apply the integration by parts formula again:

$\int e^{5x} \cos(3x) \, dx = \frac{1}{5} e^{5x} \cos(3x) - \int \frac{1}{5} e^{5x} (-3\sin(3x)) \, dx$

Simplifying:

$\int e^{5x} \cos(3x) \, dx = \frac{1}{5} e^{5x} \cos(3x) + \frac{3}{5} \int e^{5x} \sin(3x) \, dx$

Now we substitute this back into the original equation:

$\int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{5} \left( \frac{1}{5} e^{5x} \cos(3x) + \frac{3}{5} \int e^{5x} \sin(3x) \, dx \right)$

Distribute the terms:

$\int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{25} e^{5x} \cos(3x) - \frac{9}{25} \int e^{5x} \sin(3x) \, dx$

Now, collect the integrals of $e^{5x} \sin(3x)$ on one side:

$\int e^{5x} \sin(3x) \, dx + \frac{9}{25} \int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{25} e^{5x} \cos(3x)$

Factor out the integral:

$\left( 1 + \frac{9}{25} \right) \int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{25} e^{5x} \cos(3x)$

Simplify the coefficient:

$\frac{34}{25} \int e^{5x} \sin(3x) \, dx = \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{25} e^{5x} \cos(3x)$

Finally, solve for the integral:

$\int e^{5x} \sin(3x) \, dx = \frac{25}{34} \left( \frac{1}{5} e^{5x} \sin(3x) - \frac{3}{25} e^{5x} \cos(3x) \right)$

This simplifies to:

$\int e^{5x} \sin(3x) \, dx = \frac{5}{34} e^{5x} \sin(3x) - \frac{3}{34} e^{5x} \cos(3x) + C$

Where $C$ is the constant of integration.

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Would you like to go over the steps in more detail, or do you have other questions? Here are some related questions:

1. How does integration by parts work for other trigonometric-exponential integrals?
2. How would the result change if we had a different trigonometric function, like $\cos(3x)$ instead of $\sin(3x)$?
3. What happens if we apply the method of undetermined coefficients instead of integration by parts?
4. How can you recognize when to use integration by parts?
5. What is the general form for integrating products of exponentials and trigonometric functions?

Tip: When dealing with integrals involving products of exponential and trigonometric functions, remember that you often need to use integration by parts twice.