Graphing and Limits for Improper Integrals

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In summary, an improper integral is when the limits of integration are infinite or the integrand is undefined. To evaluate an improper integral, techniques like limits, comparison, or integration by parts can be used. The difference between a proper and improper integral is that a proper integral has finite limits and a continuous integrand, while an improper integral has infinite limits or a discontinuous integrand. Improper integrals are used for finding areas under curves that don't satisfy the conditions for a proper integral and are commonly used in physics and engineering. Some techniques for evaluating improper integrals include the limit definition, comparison test, and integration by parts, as well as others like partial fractions and trigonometric substitutions depending on the integral.
  • #1
Dan350
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Do you need to know how to graph in order to establish which limit of an improper integrals is going to infinity?? for example:

∫tan(3x)dx from 0 to ∏/6

The integral diverges,, but how do you figure which constant you should use
In this problem they put it as the limit a b aproaches ∏/6 form the left.
then they just simply integtrate

Thank you so much!
 
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  • #2
A full-out graph wouldn't be necessary, but knowledge of tan(x)'s behavior is useful. Plugging in the values should reveal that tan(pi/2) = 1/0 on the unit circle, this requiring an improper integral.
 

1. What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or where the integrand is undefined at some point within the interval of integration.

2. How do you evaluate an improper integral?

To evaluate an improper integral, you must first identify the type of improper integral (infinite limit or undefined integrand). Then, you can use techniques such as limits, comparison, or integration by parts to evaluate the integral.

3. What is the difference between a proper and improper integral?

A proper integral has finite limits of integration and a continuous integrand over the entire interval. An improper integral has infinite limits or a discontinuous integrand, making it more challenging to evaluate.

4. When would you use an improper integral?

Improper integrals are used to find areas under curves that do not satisfy the conditions for a proper integral, typically involving infinite intervals or discontinuous functions. They are also used in many applications in physics and engineering.

5. What are some common techniques for evaluating improper integrals?

Some common techniques for evaluating improper integrals include the limit definition of an integral, the comparison test, and the integration by parts method. Other techniques, such as partial fractions and trigonometric substitutions, may also be used depending on the specific integral.

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