Evaluate the following integral from 0 to infinity

Thread starter mrdad123
Start date Nov 11, 2010
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Infinity Integral

In summary, evaluating an integral from 0 to infinity involves finding the numerical value of the area under a curve that extends indefinitely in the positive direction on the x-axis. This type of integral is different from a regular integral because the upper limit is infinity, and its significance is that the function being integrated grows without bound in the positive direction. It is possible for an integral from 0 to infinity to have a finite value when the function being integrated approaches zero or a finite value as x approaches infinity. Real-life applications of this type of integral include calculating energy, decay, and profits over infinite distances or time periods.

Nov 11, 2010

mrdad123

Evaluate the following integral from 0 to infinity. (see attached for better picture)

e^(-ax)-e^(bx)
------------------ dx
x

Remarks:

a , b > 0
a < b

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Nov 11, 2010

hunt_mat

Homework Helper

Have you tried residue calculus?

Nov 11, 2010

mrdad123

No, we have not learned how to calculate integrals with residue calculus just yet.

Do you think it's possible with integration by parts?

FAQ: Evaluate the following integral from 0 to infinity

What does it mean to evaluate an integral from 0 to infinity?

Evaluating an integral from 0 to infinity means finding the numerical value of the area under a curve that extends indefinitely in the positive direction on the x-axis, starting from the point x=0.

How is an integral from 0 to infinity different from a regular integral?

The main difference is that the upper limit of the integral is infinity, which means the integration process continues indefinitely. This type of integral is also known as an improper integral.

3. What is the significance of the upper limit being infinity in an integral?

The upper limit being infinity means that the function being integrated grows without bound in the positive direction. This can represent situations such as unlimited growth or accumulation over time.

4. Can an integral from 0 to infinity have a finite value?

Yes, it is possible for an integral from 0 to infinity to have a finite value. This occurs when the function being integrated approaches zero or a finite value as x approaches infinity. In these cases, the area under the curve can be calculated and results in a finite value.

5. What are some real-life applications of evaluating an integral from 0 to infinity?

Some examples include calculating the total energy or work done by a force that varies over an infinite distance, determining the total amount of radioactive material that has decayed over time, and finding the total profit earned by a company over an indefinite period.