Math Problem Statement
onsider the following two integrals. I1=∫1−1dxx2I2=∫10cosxx−−√dx
Solution
The integrals given are:
Let's analyze each integral for convergence or divergence.
Integral
The integrand is undefined at , making it an improper integral. We can split this integral as follows:
Evaluating
This part of the integral is:
Calculating the integral:
As , , so the integral diverges.
Thus, diverges due to the behavior at .
Integral
This integral is also improper because approaches zero as , creating a singularity.
Rewrite as:
Since is bounded and in the denominator implies that has a singularity at , we can approximate the behavior near by bounding .
Convergence of
Consider:
As , this expression approaches , so the integral converges.
Thus, converges.
Summary
- diverges due to a singularity at .
- converges despite the singularity at .
Would you like a more detailed breakdown of any steps?
- How do improper integrals differ in convergence based on singularity types?
- Can integrals converge even with infinite discontinuities within the integration range?
- What does it mean for an integral to be conditionally convergent?
- How does the limit process help in evaluating improper integrals?
- Are there specific functions that often lead to divergent integrals?
Tip: When dealing with improper integrals, always identify any points of discontinuity and assess the behavior of the integrand around those points.
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Math Problem Analysis
Mathematical Concepts
Calculus
Improper Integrals
Convergence and Divergence
Formulas
I1 = ∫_{-1}^{1} (1/x^2) dx
I2 = ∫_{0}^{1} (cos(x)/√x) dx
Theorems
Divergence and Convergence of Improper Integrals
Suitable Grade Level
Undergraduate Calculus
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