The given integral is:

$\int x^3 \cdot e^x \, dx$

To solve this, we can use the technique called Integration by Parts, which follows the formula:

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

Here, we'll choose:

• $u = x^3 \Rightarrow du = 3x^2 \, dx$
• $dv = e^x \, dx \Rightarrow v = e^x$

Plugging into the formula:

$\int x^3 \cdot e^x \, dx = x^3 e^x - \int 3x^2 e^x \, dx$

Now, repeat the process for $\int 3x^2 e^x \, dx$:

• $u = x^2 \Rightarrow du = 2x \, dx$
• $dv = e^x \, dx \Rightarrow v = e^x$

$\int 3x^2 e^x \, dx = 3(x^2 e^x - \int 2x e^x \, dx)$

For $\int 2x e^x \, dx$:

• $u = x \Rightarrow du = 1 \, dx$
• $dv = e^x \, dx \Rightarrow v = e^x$

$\int 2x e^x \, dx = 2(x e^x - \int e^x \, dx) = 2(x e^x - e^x)$

Now we back-substitute:

$\int x^3 \cdot e^x \, dx = x^3 e^x - 3(x^2 e^x - 2(x e^x - e^x)) + C$

Simplifying:

$\int x^3 \cdot e^x \, dx = e^x (x^3 - 3x^2 + 6x - 6) + C$

where $C$ is the constant of integration.

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Do you want details on any step, or have any questions? Here are 5 related problems:

1. Calculate $\int x^2 e^x \, dx$.
2. Evaluate $\int x^3 e^{-x} \, dx$.
3. Find $\int x^4 e^x \, dx$.
4. Use integration by parts to solve $\int x \sin(x) \, dx$.
5. Determine $\int x^3 e^x \, dx$ using a tabular method.

Tip: For repeated integration by parts, consider the tabular method—it can simplify the process significantly!