Math Problem Statement
Solution
The integral you provided is:
with the substitution .
Let's solve this step by step using the substitution method:
Step 1: Substitution
Given: Differentiate with respect to to find :
This implies:
Step 2: Substitute into the Integral
Now substitute into the integral:
This simplifies to:
Step 3: Integrate
The integral of with respect to is simply :
Step 4: Substitute Back
Finally, substitute back :
Thus, the solution to the integral is:
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:
- How does the substitution method work in integrals?
- What are the common substitutions used for Gaussian integrals?
- Can you explain how integration by parts might apply to a similar problem?
- What are other techniques to solve integrals involving exponentials and polynomials?
- How do definite integrals differ when using substitution?
Tip: Always check if substitution simplifies the integral by directly transforming the differential 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