https://r2.erweima.ai/mathbot/image/1725568257707ep0g02tk.jpg

The image shows an integral of the form:

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

This is a problem where we can solve using substitution. Let's go through the solution:

1. **Substitute** \( u = x^2 \):
   - Then, \( du = 2x \, dx \) or equivalently, \( \frac{du}{2} = x \, dx \).
   
2. Rewrite the integral:
   \[
   \int x^3 e^{x^2} \, dx = \int x^2 \cdot x e^{x^2} \, dx
   \]
   Now use the substitution \( u = x^2 \), so the integral becomes:
   \[
   \int x^2 e^u \cdot \frac{du}{2x}
   \]
   This simplifies to:
   \[
   \frac{1}{2} \int u e^u \, du
   \]

3. Apply integration by parts:
   Let \( v = u \) and \( dw = e^u \, du \), so that \( dv = du \) and \( w = e^u \). Now, the integration by parts formula \( \int v \, dw = v \cdot w - \int w \, dv \) gives:
   \[
   \frac{1}{2} \left( u e^u - \int e^u \, du \right)
   \]
   Simplifying the integral:
   \[
   \frac{1}{2} \left( u e^u - e^u \right) + C
   \]
   Substituting back \( u = x^2 \):
   \[
   \frac{1}{2} \left( x^2 e^{x^2} - e^{x^2} \right) + C
   \]
   This simplifies to:
   \[
   \frac{1}{2} e^{x^2} (x^2 - 1) + C
   \]

So, the correct answer is:

\[
\boxed{\frac{e^{x^2}}{2} (x^2 - 1) + C}
\]

The correct option based on this result is option **C**.

---

Do you want further details or have any questions?  
Here are 5 related questions to expand on this topic:

1. How do we apply integration by parts in general?
2. What other problems can be solved using substitution?
3. How would you solve the integral without substitution?
4. Can the answer change with different boundary conditions?
5. What is the significance of the constant of integration \(C\)?

**Tip:** Integration by parts is especially useful when you encounter a product of functions like \( u \cdot dv \), and one part becomes simpler when differentiated!

Learn how to solve the integral of x^3 e^(x^2) using substitution and integration by parts. This calculus problem explains step-by-step how to handle integrals involving exponential functions and powers of x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation