The integral shown in the image is:

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

To solve this, we can use integration by parts, where:

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

Step-by-Step Solution

1. Choose $u$ and $dv$:

   • Let $u = x^3$ (this choice makes $du$ simpler as we differentiate a polynomial).
   • Let $dv = e^x \, dx$ (since $e^x$ is straightforward to integrate).

2. Differentiate $u$ and integrate $dv$:

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

3. Apply the integration by parts formula: $\int x^3 e^x \, dx = u v - \int v \, du$ Substitute $u = x^3$, $v = e^x$, and $du = 3x^2 \, dx$: $= x^3 e^x - \int e^x \cdot 3x^2 \, dx$ $= x^3 e^x - 3 \int x^2 e^x \, dx$

4. Repeat integration by parts on $\int x^2 e^x \, dx$:

   • Let $u = x^2$, $dv = e^x \, dx$.
   • Differentiate and integrate again:
     • $u = x^2 \Rightarrow du = 2x \, dx$
     • $dv = e^x \, dx \Rightarrow v = e^x$
   • Apply integration by parts: $\int x^2 e^x \, dx = x^2 e^x - \int e^x \cdot 2x \, dx$ $= x^2 e^x - 2 \int x e^x \, dx$

5. Repeat one more time for $\int x e^x \, dx$:

   • Let $u = x$, $dv = e^x \, dx$.
   • Differentiate and integrate:
     • $u = x \Rightarrow du = dx$
     • $dv = e^x \, dx \Rightarrow v = e^x$
   • Apply integration by parts: $\int x e^x \, dx = x e^x - \int e^x \, dx$ $= x e^x - e^x$

6. Combine all results:

   Substitute back each part until we obtain the full expression for the integral. This will give:

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

Would you like a detailed combination of all the steps in the final expression?

Related Questions:

1. How does integration by parts work for polynomial functions multiplied by exponentials?
2. Can integration by parts be applied to trigonometric functions as well?
3. Why does repeated application of integration by parts eventually solve this integral?
4. What other methods can be used to integrate expressions like $x^n e^x$?
5. How does the choice of $u$ and $dv$ affect the ease of solving an integral?

Tip: When using integration by parts, try to choose $u$ as a function that simplifies when differentiated, such as polynomials.