The integral you provided is:

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

To solve this, we use integration by parts multiple times. Integration by parts is based on the formula:

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

Step 1: Choose $u$ and $dv$

Let:

• $u = x^3$, so $du = 3x^2 \, dx$
• $dv = e^{2x} \, dx$, so $v = \frac{1}{2}e^{2x}$ (since the derivative of $e^{2x}$ is $2e^{2x}$)

Step 2: Apply Integration by Parts

Now apply the integration by parts formula:

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

Simplify:

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

Step 3: Repeat the process for $\int x^2 e^{2x} \, dx$

Let:

• $u = x^2$, so $du = 2x \, dx$
• $dv = e^{2x} \, dx$, so $v = \frac{1}{2}e^{2x}$

Now apply integration by parts again:

$\int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \int \frac{1}{2} 2x e^{2x} \, dx$

Simplify:

$= \frac{1}{2} x^2 e^{2x} - \frac{1}{2} \int 2x e^{2x} \, dx$

Step 4: Repeat for $\int x e^{2x} \, dx$

Let:

• $u = x$, so $du = dx$
• $dv = e^{2x} \, dx$, so $v = \frac{1}{2}e^{2x}$

Now apply integration by parts:

$\int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} \, dx$

The last integral is straightforward:

$\int e^{2x} \, dx = \frac{1}{2}e^{2x}$

So:

$\int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{4}e^{2x}$

Step 5: Combine all results

Now substitute back into the original equation:

$\int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} \left( \frac{1}{2} x e^{2x} - \frac{1}{4}e^{2x} \right)$

Simplify:

$= \frac{1}{2} x^2 e^{2x} - \frac{1}{4} x e^{2x} + \frac{1}{8}e^{2x}$

Finally, substitute this back into the previous result:

$\int x^3 e^{2x} \, dx = \frac{1}{2} x^3 e^{2x} - \frac{3}{2} \left( \frac{1}{2} x^2 e^{2x} - \frac{1}{4} x e^{2x} + \frac{1}{8}e^{2x} \right)$

Simplify:

$\int x^3 e^{2x} \, dx = \frac{1}{2} x^3 e^{2x} - \frac{3}{4} x^2 e^{2x} + \frac{3}{8} x e^{2x} - \frac{3}{16}e^{2x} + C$

Final Answer:

$\int x^3 e^{2x} \, dx = \frac{1}{2} x^3 e^{2x} - \frac{3}{4} x^2 e^{2x} + \frac{3}{8} x e^{2x} - \frac{3}{16}e^{2x} + C$

Would you like any further details or clarification on any step?

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Here are 5 related questions:

1. How do you choose the $u$ and $dv$ in integration by parts?
2. Can integration by parts be used for other transcendental functions like $\sin(x)$?
3. What is the general formula for integration by parts?
4. How does repeated integration by parts work in practical applications?
5. What are the conditions under which integration by parts simplifies a problem?

Tip: When integrating by parts multiple times, always check for a recursive pattern to simplify calculations.