Math Problem Statement

Evaluate the integral: ∫ xe^(-x^2) dx, using u = -x^2

Solution

The integral you provided is:

xex2dx\int x e^{-x^2} \, dx

with the substitution u=x2u = -x^2.

Let's solve this step by step using the substitution method:

Step 1: Substitution

Given: u=x2u = -x^2 Differentiate uu with respect to xx to find dudu: du=2xdxdu = -2x \, dx

This implies: du2=xdx\frac{du}{-2} = x \, dx

Step 2: Substitute into the Integral

Now substitute into the integral: xex2dx=eudu2\int x e^{-x^2} \, dx = \int e^u \cdot \frac{du}{-2}

This simplifies to: 12eudu-\frac{1}{2} \int e^u \, du

Step 3: Integrate

The integral of eue^u with respect to uu is simply eue^u: 12eudu=12eu+C-\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C

Step 4: Substitute Back

Finally, substitute back u=x2u = -x^2: 12eu+C=12ex2+C-\frac{1}{2} e^u + C = -\frac{1}{2} e^{-x^2} + C

Thus, the solution to the integral is: xex2dx=12ex2+C\int x e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2} + C

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions you might find interesting:

  1. How does the substitution method work in integrals?
  2. What are the common substitutions used for Gaussian integrals?
  3. Can you explain how integration by parts might apply to a similar problem?
  4. What are other techniques to solve integrals involving exponentials and polynomials?
  5. How do definite integrals differ when using substitution?

Tip: Always check if substitution simplifies the integral by directly transforming the differential dxdx and integrand.

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method

Formulas

Substitution: u = -x^2
Integral of e^u

Theorems

Fundamental Theorem of Calculus
Substitution in Integration

Suitable Grade Level

Undergraduate Calculus