Math Problem Statement
Solution
Let’s break down and solve each part of this exercise step by step.
1. (a) Show that is convergent and calculate its value.
Solution:
The integral is:
The general formula for the integral of is:
Evaluate the definite integral:
- As , .
- At , .
Thus:
The value is:
1. (b) Deduce that is convergent.
Solution:
The function decreases faster than as . Since converges, by comparison, also converges by the comparison test.
2. (a) Show that is convergent and calculate its value.
Solution:
The integral is:
Using integration by parts:
Evaluate the definite integral:
-
At :
-
At :
Thus:
The value is:
2. (b) Show that is convergent and calculate its value.
Solution:
Substitute , so , and . The limits transform as:
- When , .
- When , .
The integral becomes:
Simplify:
Use the identity :
-
Compute :
-
Compute : Evaluate from to :
Thus:
The value is:
Summary of Results:
- (a) .
- (b) is convergent.
- (a) .
- (b) .
Would you like further details on any step or explanation? Here are some related questions to consider:
- How does the comparison test for convergence work?
- What are the conditions for applying integration by parts?
- Why does the substitution simplify integrals involving ?
- Can be integrated analytically over ?
- How does L'Hopital's Rule apply to limits like ?
Tip: Always check for the behavior of the integrand at the boundaries to assess convergence!
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Math Problem Analysis
Mathematical Concepts
Convergence of improper integrals
Exponential functions
Natural logarithm integration
Trigonometric substitution in integrals
Formulas
\(\int e^{-x} \, dx = -e^{-x} + C\)
Integration by parts: \(\int u \, dv = uv - \int v \, du\)
Trigonometric identities: \(\sin^2 t = \frac{1 - \cos(2t)}{2}\)
Theorems
Comparison Test for convergence
L'Hopital's Rule for evaluating limits
Suitable Grade Level
Undergraduate calculus
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