Math Problem Statement

∫x^3*e^x^2 dx

Solution

To solve the integral x3ex2dx\int x^3 e^{x^2} \, dx, we can use the method of substitution.

Step 1: Substitution

Let: u=x2u = x^2 Then, differentiate both sides with respect to xx: du=2xdxordu2=xdxdu = 2x \, dx \quad \text{or} \quad \frac{du}{2} = x \, dx

Step 2: Rewrite the integral

Now, rewrite the original integral in terms of uu. Since x3=xx2=xux^3 = x \cdot x^2 = x \cdot u, we substitute u=x2u = x^2 and xdx=du2x \, dx = \frac{du}{2}: x3ex2dx=xueudu2\int x^3 e^{x^2} \, dx = \int x \cdot u \cdot e^u \cdot \frac{du}{2} =12ueudu= \frac{1}{2} \int u e^u \, du

Step 3: Use integration by parts

Now, apply integration by parts. Let: v=uanddw=euduv = u \quad \text{and} \quad dw = e^u \, du Differentiate and integrate: dv=duandw=eudv = du \quad \text{and} \quad w = e^u Now, apply the formula for integration by parts: vdw=vwwdv\int v \, dw = v \cdot w - \int w \, dv Substituting the values: 12ueudu=12(ueueudu)\frac{1}{2} \int u e^u \, du = \frac{1}{2} \left( u e^u - \int e^u \, du \right) =12(ueueu)=12eu(u1)= \frac{1}{2} \left( u e^u - e^u \right) = \frac{1}{2} e^u (u - 1)

Step 4: Substitute back in terms of xx

Recall that u=x2u = x^2, so: 12eu(u1)=12ex2(x21)\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: x3ex2dx=12ex2(x21)+C\int x^3 e^{x^2} \, dx = \frac{1}{2} e^{x^2} (x^2 - 1) + C where CC 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 xex2dx\int x e^{x^2} \, dx?
  2. Can you apply integration by parts to x2exdx\int x^2 e^x \, dx?
  3. What is the derivative of 12ex2(x21)\frac{1}{2} e^{x^2} (x^2 - 1)?
  4. How do you handle integrals involving ex2e^{x^2} without a polynomial?
  5. What is the solution to ex2dx\int e^{x^2} \, dx?

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

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method
Integration by Parts
Exponential Functions

Formulas

Substitution: u = x^2, du = 2x dx
Integration by Parts: ∫v dw = v * w - ∫w dv

Theorems

Integration by Parts Theorem
Substitution Rule in Integration

Suitable Grade Level

Undergraduate Level (Calculus II)