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The integral given is:

\[
\int x^3 e^{2x} \, dx
\]

To solve this integral, we can use the method of integration by parts. The formula for integration by parts is:

\[
\int u \, dv = uv - \int v \, du
\]

For this integral, we can let:
- \( u = x^3 \), which means \( du = 3x^2 \, dx \)
- \( dv = e^{2x} \, dx \), which means \( v = \frac{1}{2} e^{2x} \) (since the integral of \( e^{2x} \) with respect to \( x \) is \( \frac{1}{2} e^{2x} \))

Now, apply the integration by parts formula:

\[
\int x^3 e^{2x} \, dx = x^3 \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 3x^2 \, dx
\]

Simplify:

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

Next, we need to integrate \( \int x^2 e^{2x} \, dx \). We can apply integration by parts again, this time setting:
- \( u = x^2 \), so \( du = 2x \, dx \)
- \( dv = e^{2x} \, dx \), so \( v = \frac{1}{2} e^{2x} \)

Applying integration by parts again:

\[
\int x^2 e^{2x} \, dx = x^2 \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 2x \, dx
\]

Simplify:

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

For the remaining integral \( \int x e^{2x} \, dx \), apply integration by parts one more time:
- \( u = x \), so \( du = dx \)
- \( dv = e^{2x} \, dx \), so \( v = \frac{1}{2} e^{2x} \)

Thus:

\[
\int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx
\]

The integral of \( \frac{1}{2} e^{2x} \) is \( \frac{1}{4} e^{2x} \), so:

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

Now, substituting back:

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

Substituting back again:

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

Finally, include the constant of integration \( C \):

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

This is the integral of the given expression. Would you like more details or have any questions?

Here are five related questions for practice:
1. How would you integrate \( \int x^2 e^{x} \, dx \)?
2. What is the integral \( \int x^4 e^{3x} \, dx \) using the same method?
3. How does integration by parts differ from substitution?
4. What are the key indicators that integration by parts is the appropriate method?
5. How can you check your answer for an integral involving exponential functions?

**Tip:** When applying integration by parts multiple times, it's helpful to keep track of each step carefully to avoid mistakes.

Learn how to solve the integral ∫ x^3 e^{2x} dx step-by-step using integration by parts. This problem involves advanced calculus techniques and exponential functions. Suitable for advanced college-level students and anyone interested in calculus.

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