Math Problem Statement
Evaluate the integral \( \int x^2 e^x \, dx \)
Solution
To evaluate the integral
we can use the technique of integration by parts multiple times. Let’s proceed with the steps:
Step 1: Integration by Parts (First Application)
For integration by parts, we use the formula:
Choose:
- (which means )
- (which gives )
Then,
Step 2: Integration by Parts (Second Application)
Now, we need to evaluate the integral .
Choose:
- (which means )
- (which gives )
Then,
Step 3: Final Integration
Now we integrate , which is straightforward:
Putting it All Together
Substituting back, we get:
Simplifying, we find:
where is the constant of integration.
Would you like more details, or do you have any questions?
Related Questions
- How does integration by parts work, and when is it useful?
- What are alternative methods for evaluating integrals like this one?
- How would the answer change if the integral was for any integer ?
- Could substitution help simplify similar integrals?
- How can we confirm the result by differentiating the answer?
Tip
When using integration by parts repeatedly, keep track of terms carefully to avoid errors in substitution.
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Math Problem Analysis
Mathematical Concepts
Integration
Integration by Parts
Exponential Functions
Formulas
Integration by parts formula: \( \int u \, dv = uv - \int v \, du \)
Theorems
Integration by Parts
Suitable Grade Level
Undergraduate Calculus