Questions: Evaluate the integral ∫ x^3 · e^x dx using tabular integration.

Evaluate the integral \( \int x^{3} \cdot e^{x} \, dx \).

Identify the functions for integration by parts.

Let \( u = x^{3} \) and \( dv = e^{x} \, dx \).

Differentiate \( u \) and integrate \( dv \).

Then, \( du = 3x^{2} \, dx \) and \( v = e^{x} \).

Apply the integration by parts formula.

Using the formula \( \int u \, dv = uv - \int v \, du \), we have: \[ \int x^{3} e^{x} \, dx = x^{3} e^{x} - \int e^{x} (3x^{2}) \, dx \]

Repeat the integration by parts for \( \int 3x^{2} e^{x} \, dx \).

Let \( u = 3x^{2} \) and \( dv = e^{x} \, dx \), leading to: \[ \int 3x^{2} e^{x} \, dx = 3x^{2} e^{x} - \int e^{x} (6x) \, dx \]

Continue this process for \( \int 6x e^{x} \, dx \).

Let \( u = 6x \) and \( dv = e^{x} \, dx \), resulting in: \[ \int 6x e^{x} \, dx = 6x e^{x} - \int e^{x} (6) \, dx \]

Evaluate the final integral.

The integral \( \int 6 e^{x} \, dx = 6 e^{x} \).

Combine all parts to find the final result.

Putting it all together, we have: \[ \int x^{3} e^{x} \, dx = (x^{3} - 3x^{2} + 6x - 6)e^{x} + C \]

The final answer is \( \int x^{3} \cdot e^{x} \, dx = (x^{3} - 3x^{2} + 6x - 6)e^{x} + C \).

The final answer is \( \boxed{(x^{3} - 3x^{2} + 6x - 6)e^{x} + C} \).