- #1
ozgunozgur
- 27
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Can Improper Integrals Help Solve This Inequality?
- MHB
- Thread starter ozgunozgur
- Start date
In summary, an improper integral is an integral with either infinite limits of integration or an unbounded integrand within the interval of integration. A proper integral, on the other hand, has finite limits of integration and a bounded integrand. An integral can be determined as improper if it meets any of the following conditions: infinite interval of integration, unbounded integrand, or a discontinuity within the interval of integration. The two types of improper integrals are type 1 and type 2, with type 1 having infinite limits of integration and type 2 having an unbounded integrand or discontinuity within the interval of integration. To evaluate an improper integral, one can use the limit definition, split the integral, or use a comparison test
- #2
skeeter
- 1,103
- 1
note ...
$\dfrac{x\sqrt{x}}{x^2-1} > \dfrac{1}{\sqrt{x}}$
and $\displaystyle \int_2^\infty \dfrac{dx}{\sqrt{x}}$ is divergent.
$\dfrac{x\sqrt{x}}{x^2-1} > \dfrac{1}{\sqrt{x}}$
and $\displaystyle \int_2^\infty \dfrac{dx}{\sqrt{x}}$ is divergent.
FAQ: Can Improper Integrals Help Solve This Inequality?
What is an improper integral?
An improper integral is an integral that either has infinite limits of integration or the integrand is undefined at some point within the limits of integration.
How do you determine if an improper integral converges or diverges?
To determine if an improper integral converges or diverges, you can use the limit comparison test, the integral comparison test, or the p-test. If the limit of the integral approaches a finite number as the limits of integration approach infinity, then the integral converges. If the limit approaches infinity or negative infinity, then the integral diverges.
Can an improper integral have both infinite limits of integration and an undefined integrand?
Yes, an improper integral can have both infinite limits of integration and an undefined integrand. In this case, the integral is considered to be doubly improper.
How do you evaluate an improper integral?
To evaluate an improper integral, you can use the definition of the improper integral, which involves taking the limit as the upper and lower limits of integration approach infinity or negative infinity. You can also use techniques such as integration by parts or substitution to evaluate the integral.
Are there any real-world applications of improper integrals?
Yes, improper integrals have many real-world applications, such as in physics, engineering, and economics. For example, they can be used to calculate the total distance traveled by an object with changing velocity, the total work done by a variable force, or the total profit or loss in a business with changing revenue and expenses.