Improper Integrals - Infinite Intervals

In summary, the conversation discusses the evaluation of an improper integral with an infinite interval. The person asking for help is unsure of how to properly evaluate the integral, but eventually realizes that substitution with u=ln(x) is the correct approach. This leads to a successful solution of the integral.
  • #1
forestmine
203
0
Improper Integrals -- Infinite Intervals

Homework Statement



Evaluate the integral.

(from e to infinity) ∫(25/x(lnx)^3)dx

Homework Equations




The Attempt at a Solution



I know that for evaluating improper integrals, you can take the limit as t approaches infinity of the given integral, but my problem is in evaluating the integral itself.

25 comes out of the integrand, and so I'm left with dx/x(lnx)^3, and this is where I'm confused. I'm guessing integration by parts is necessary to properly evaluate this, but I'm not entirely sure. I tried setting u = (lnx)^-3, which gives me du=-3lnx/x*dx, but then I'm left with dv=xdx, so v=1/2x^2, but based on the solution, I don't believe I'm at all headed in the right direction.

Any help would be greatly appreciated!
 
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  • #2


forestmine said:
I'm guessing integration by parts is necessary to properly evaluate this,


Use the substitution u=ln(x) instead.


ehild
 
  • #3


I did exactly that and it works out perfectly.

Thanks a lot!
 

1. What is an improper integral?

An improper integral is an integral where either the upper or lower limit of integration is infinite or the integrand has a vertical asymptote within the interval of integration. This means that the area under the curve cannot be calculated using traditional methods and requires a special approach.

2. How do you evaluate an improper integral?

To evaluate an improper integral, you must first break it into smaller intervals where the integrand is continuous. Then, you can evaluate each interval individually and take the limit as the intervals approach infinity or the vertical asymptote. If the limit exists, it is the value of the improper integral.

3. What are the types of improper integrals?

There are two types of improper integrals: Type I and Type II. Type I improper integrals have an infinite limit of integration, while Type II improper integrals have a vertical asymptote within the interval of integration.

4. What is the convergence test for improper integrals?

The convergence test for improper integrals states that if the limit of the integral exists, then the improper integral is convergent. If the limit does not exist, the improper integral is divergent.

5. Can an improper integral have both infinite limits of integration and a vertical asymptote?

Yes, an improper integral can have both infinite limits of integration and a vertical asymptote within the interval of integration. This would be considered a Type II improper integral and would require a special approach to evaluate.

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