To solve the integral $\int x^3 e^{x^2} \, dx$, we can use the method of substitution.

Step 1: Substitution

Let: $u = x^2$ Then, differentiate both sides with respect to $x$: $du = 2x \, dx \quad \text{or} \quad \frac{du}{2} = x \, dx$

Step 2: Rewrite the integral

Now, rewrite the original integral in terms of $u$. Since $x^3 = x \cdot x^2 = x \cdot u$, we substitute $u = x^2$ and $x \, dx = \frac{du}{2}$: $\int x^3 e^{x^2} \, dx = \int x \cdot u \cdot e^u \cdot \frac{du}{2}$ $= \frac{1}{2} \int u e^u \, du$

Step 3: Use integration by parts

Now, apply integration by parts. Let: $v = u \quad \text{and} \quad dw = e^u \, du$ Differentiate and integrate: $dv = du \quad \text{and} \quad w = e^u$ Now, apply the formula for integration by parts: $\int v \, dw = v \cdot w - \int w \, dv$ Substituting the values: $\frac{1}{2} \int u e^u \, du = \frac{1}{2} \left( u e^u - \int e^u \, du \right)$ $= \frac{1}{2} \left( u e^u - e^u \right) = \frac{1}{2} e^u (u - 1)$

Step 4: Substitute back in terms of $x$

Recall that $u = x^2$, so: $\frac{1}{2} e^u (u - 1) = \frac{1}{2} e^{x^2} (x^2 - 1)$

Final Answer:

Thus, the solution to the integral is: $\int x^3 e^{x^2} \, dx = \frac{1}{2} e^{x^2} (x^2 - 1) + C$ where $C$ is the constant of integration.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. How would you solve $\int x e^{x^2} \, dx$?
2. Can you apply integration by parts to $\int x^2 e^x \, dx$?
3. What is the derivative of $\frac{1}{2} e^{x^2} (x^2 - 1)$?
4. How do you handle integrals involving $e^{x^2}$ without a polynomial?
5. What is the solution to $\int e^{x^2} \, dx$?

Tip: When using substitution, always check if the new variables simplify the integral and avoid unnecessary complexity.