How Do You Solve This Improper Integral with e^(t*(b-s))?

In summary, the conversation is about solving an improper integral \int e^{t*(b-s)} evaluated from 0 to \infty. The correct solution is 1/(b-s), but the person initially made a mistake by using a negative sign in the exponent. They also learned how to write equations using LaTeX.
  • #1
Rib5
59
0
Hey guys, I was doing some homework problems and I ran into a problem regarding how to solve a certain improper integral.

[tex]\int e^{t*(b-s)}[/tex] evaluated from 0 to [tex]\infty[/tex]

So I take the integral and get

[tex]\frac{\int e^{t*(b-s)}}{-(b-s)}[/tex] which evaluated from 0 to [tex]\infty[/tex]

gives me 0 - [tex]\frac{1}{-(b-s)}[/tex]

which is 1/(b-s)

The answer should be 1/(s-b). Can anyone help me figure out what I am messing up?
 
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  • #2
Where does the minus in your second step come from?
 
  • #3
Rib5 said:
Hey guys, I was doing some homework problems and I ran into a problem regarding how to solve a certain improper integral.

[tex]\int e^{t*(b-s)}[/tex] evaluated from 0 to [tex]\infty[/tex]

Click on the image below to see how to write this a little nicer with LaTeX:

[tex]\int_{0}^{\infty} e^{(b-s)t}dt[/tex]

Is this what you meant? (you didn't actually specify which variable you are integrating over)

So I take the integral and get

[tex]\frac{\int e^{(b-s)t}}{-(b-s)}[/tex] which evaluated from 0 to [tex]\infty[/tex]

Surely you mean

[tex]\int_{0}^{\infty} e^{t*(b-s)}dt= \frac{e^{(b-s)t}}{(b-s)} {\left|}_{0}^{\infty}[/tex]

right?

Also, are you told that [itex](b-s)<0[/itex]? If not, you will need to examine two different cases.
 
  • #4
Thanks guys, I feel really stupid now. Earlier today I did a bunch of integrals where the sign on the power was negative and I think I ended up mixing up the what the integral of [tex]e^{at}[/tex] is.

Also thanks for the tip about Latex
 

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 function being integrated is undefined at one or more points within the interval of integration.

2. How do you solve a simple improper integral?

To solve a simple improper integral, you must first determine if it is convergent or divergent. If it is convergent, you can solve it by using the limit definition of integration or by using integration techniques such as substitution or integration by parts.

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

A definite integral has finite limits of integration and can be solved using standard integration techniques. An improper integral has either infinite limits of integration or an undefined function within the interval of integration, requiring special techniques to solve.

4. Can improper integrals have negative values?

Yes, improper integrals can have negative values if the function being integrated is negative over the interval of integration or if the limits of integration are negative.

5. Why do we need to use improper integrals?

Improper integrals are necessary to solve integrals of functions that are not continuous over the entire interval of integration or have infinite limits of integration. They also allow us to find the area under curves that extend to infinity or have vertical asymptotes.

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