Math Problem Statement
∫∞ 1 𝑥𝑒−𝑥𝑑𝑥
Solution
The given integral is:
To solve this, we can use integration by parts.
Let:
- , so
- , so
Now applying the integration by parts formula:
Substitute the values:
Step 1: Evaluate
At :
At :
Thus:
Step 2: Solve the remaining integral
At :
At :
Thus:
Final Answer:
Now add the two results together:
Thus, the value of the integral is:
Would you like further clarification on any of the steps?
Here are some related questions:
- What are other examples of integrals solved by integration by parts?
- How do we handle improper integrals like this one in general?
- Can we apply different techniques to evaluate integrals involving ?
- What happens if we change the limits of integration?
- How does the convergence of affect integrals on an infinite interval?
Tip: Always check the behavior of terms at infinity when evaluating improper integrals.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Improper Integrals
Integration by Parts
Exponential Decay
Formulas
∫ u dv = uv - ∫ v du (Integration by Parts Formula)
∫_1^∞ e^(-x) dx = [ -e^(-x) ]_1^∞
Theorems
Integration by Parts
Behavior of Exponential Functions at Infinity
Suitable Grade Level
Undergraduate Calculus