Integral with exponential terms?

The integral in question has a solution in terms of the Gaussian hypergeometric function, but it can also be expressed using the beta function or an infinite series. These three forms are closely related and none is any more simple than another.
  • #1
fchopin
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I am doing some analysis and I have come up with the following integral:

[itex]\int \frac{e^{-ax}}{1+be^{-cx}}dx[/itex]

where [itex]a>0[/itex], [itex]b>0[/itex] and [itex]c>0[/itex].

I have found out this integral has a solution in terms of the Gaussian hypergeometric function http://en.wikipedia.org/wiki/Hypergeometric_function but it looks to me that it should have a solution in terms of simple mathematical functions.

Is there any solution to this integral in terms of simple functions?

Thanks in advance
 
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  • #2
The hypergeometric function is a simple mathematical function. You could also use the beta function or an infinite series. Those three forms are closely related and none is really any more simple than another.
 
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FAQ: Integral with exponential terms?

1. What is an integral with exponential terms?

An integral with exponential terms is a type of mathematical expression that involves both an integral (which calculates the area under a curve) and exponential functions (which involve a variable raised to a power).

2. How do you solve an integral with exponential terms?

Solving an integral with exponential terms involves using integration techniques, such as u-substitution or integration by parts, to simplify the expression and then evaluating the integral using the fundamental theorem of calculus. It may also require the use of trigonometric or logarithmic identities.

3. What is the purpose of integrals with exponential terms?

Integrals with exponential terms are used to model and solve various real-world problems in fields such as physics, engineering, and economics. They allow us to calculate important quantities such as growth rates, decay rates, and areas under curves.

4. Are there any special properties of integrals with exponential terms?

Yes, there are several special properties of integrals with exponential terms, including the power rule for integrals, which states that the integral of x^n is (x^(n+1))/(n+1), and the exponential rule for integrals, which states that the integral of e^x is e^x + C.

5. Can integrals with exponential terms be solved using software or calculators?

Yes, integrals with exponential terms can be solved using software or calculators, such as WolframAlpha or the TI-84 graphing calculator. However, it is important to understand the concepts and techniques behind solving these integrals in order to use these tools effectively.

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