Let's solve the following random integral:

$\int e^{2x} \sin(3x) \, dx$

To solve this integral, we will use integration by parts twice. Integration by parts formula is given by:

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

First, let's set: $u = e^{2x} \quad \text{and} \quad dv = \sin(3x) \, dx$

Next, we find $du$ and $v$:

$du = 2e^{2x} \, dx$

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

Now, apply the integration by parts formula:

$\int e^{2x} \sin(3x) \, dx = e^{2x} \left(-\frac{1}{3} \cos(3x)\right) - \int \left(-\frac{1}{3} \cos(3x)\right) \cdot 2e^{2x} \, dx$

Simplify:

$= -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{3} \int e^{2x} \cos(3x) \, dx$

We now need to solve the integral $\int e^{2x} \cos(3x) \, dx$. Again, we use integration by parts, setting:

$u = e^{2x} \quad \text{and} \quad dv = \cos(3x) \, dx$

Next, we find $du$ and $v$:

$du = 2e^{2x} \, dx$

$v = \int \cos(3x) \, dx = \frac{1}{3} \sin(3x)$

Now, apply the integration by parts formula again:

$\int e^{2x} \cos(3x) \, dx = e^{2x} \left(\frac{1}{3} \sin(3x)\right) - \int \left(\frac{1}{3} \sin(3x)\right) \cdot 2e^{2x} \, dx$

Simplify:

$= \frac{1}{3} e^{2x} \sin(3x) - \frac{2}{3} \int e^{2x} \sin(3x) \, dx$

Now we have:

$\int e^{2x} \cos(3x) \, dx = \frac{1}{3} e^{2x} \sin(3x) - \frac{2}{3} \int e^{2x} \sin(3x) \, dx$

Let's denote $I = \int e^{2x} \sin(3x) \, dx$. Then:

$I = -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{3} \left( \frac{1}{3} e^{2x} \sin(3x) - \frac{2}{3} I \right)$

Simplify:

$I = -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{9} e^{2x} \sin(3x) - \frac{4}{9} I$

Combine like terms:

$I + \frac{4}{9} I = -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{9} e^{2x} \sin(3x)$

$\frac{13}{9} I = -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{9} e^{2x} \sin(3x)$

Solve for $I$:

$I = \frac{9}{13} \left( -\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{9} e^{2x} \sin(3x) \right)$

Simplify:

$I = -\frac{3}{13} e^{2x} \cos(3x) + \frac{2}{13} e^{2x} \sin(3x) + C$

So, the solution to the integral is:

$\int e^{2x} \sin(3x) \, dx = -\frac{3}{13} e^{2x} \cos(3x) + \frac{2}{13} e^{2x} \sin(3x) + C$

Would you like more details or have any questions?

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1. What is the integration by parts formula?
2. How do you choose $u$ and $dv$ in integration by parts?
3. What is the process for finding $du$ and $v$ in integration by parts?
4. How do you handle integrals involving products of exponential and trigonometric functions?
5. Why did we need to use integration by parts twice in this problem?
6. How can we simplify expressions involving integrals during integration by parts?
7. What role do constants of integration ($C$) play in indefinite integrals?
8. Can the integration by parts method be applied to other types of integrals?

Tip: When solving integrals involving products of functions, always consider if integration by parts can simplify the process, especially when the integral reappears in the equation.