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eyesontheball1
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How would one go about computing the following improper integral, with limits of integration [0,∞) using residues?
[itex]\int exp(x+1/x)/x[/itex]
[itex]\int exp(x+1/x)/x[/itex]
For positive ##x##, ##\exp(x + 1/x) \geq \exp(x)##, so the integrand is bounded below by ##e^x/x##. The latter function diverges to infinity as ##x \rightarrow \infty##, so certainly it can't have finite area.eyesontheball1 said:How does it not converge though?
An improper integral is an integral where either the upper or lower limit of integration is infinite, or where the function being integrated is undefined at one or more points within the interval of integration. These types of integrals require a different approach than regular integrals in order to be solved.
An integral is improper if at least one of the following conditions is met:
A convergent improper integral is one where the limit of the integral exists and is a finite number. This means that the area under the curve can be accurately calculated. In contrast, a divergent improper integral is one where the limit of the integral does not exist or is infinite. This means that the area under the curve cannot be accurately calculated.
To solve an improper integral, you must first identify the type of improper integral it is (Type 1, Type 2, or Type 3). Then, you must use the appropriate method to solve it. This may involve finding the limit of the integral, using a substitution or a change of variables, or breaking the integral into smaller integrals. It is important to note that improper integrals may not always have a closed-form solution and may require numerical methods for approximation.
Improper integrals have various applications in physics, engineering, and economics. Some examples include calculating the center of mass of a non-uniform object, determining the total amount of radioactive material remaining in a decaying substance, and evaluating infinite series in physics and engineering problems. They can also be used to model and predict population growth and economic trends.