Integral to infinite with two variables: int_{x+1}^{infinty} [(1/4) e^{-y/4}] dy

[mihalaras]

Hello! I have the integral you can see in the attached screenshot. I am really struggling to find a proper method to solve this and I am asking for a step by step solution so I can understand the method and be able to solve similar integrals. Thank you in advance!!

 

[stapel]

Hello! I have the integral you can see in the attached screenshot. I am really struggling to find a proper method to solve this and I am asking for a step by step solution so I can understand the method and be able to solve similar integrals. Thank you in advance!!
View attachment 36314

You have loads of worked examples in your textbook and in your class notes, so one more worked example is unlikely to help. In any case, as you saw in the "Read Before Posting" message, doing your work for you is not how this site works.

If you are completely unable to get started and need all of the steps given to you, plus all the reasoning ("so [you] can understand the method and be able to solve similar integrals"), then apparently you are needing lesson instruction in addition to whatever you got in your class and your text. Unfortunately, it is not reasonably feasible to attempt, within this environment, to replicate that instruction. Fortunately, there are loads of lessons available online. (Try Googling "improper integrals infinity", without the quote-marks.)

If, once you have studied enough lessons that you're able to at least make a start on this exercise, you are unable to work your way to the answer, then please reply with a clear explanation of your thoughts and efforts so far. You first replaced the upper limit with a placeholder variable, and... then what?

Thank you!

 

[khansaheb]

@mihalaras

Can you solve:

\(\displaystyle \int{e^{-x}}dx\) = ?

 

[Steven G]

Let x = y/4. Then dx = dy/4 or dy = 4dx.
Make this substitution and then you will have the integral that khansaheb quoted above.
Post back with your work so we can help you proceed further if you need any more work.

 

[Dr.Peterson]

Hello! I have the integral you can see in the attached screenshot. I am really struggling to find a proper method to solve this and I am asking for a step by step solution so I can understand the method and be able to solve similar integrals. Thank you in advance!!
View attachment 36314

If you need an example of the steps (not for this problem, but one like it), see these textbooks, for example:

6.8: Improper Integration

When we defined the definite integral, we made two stipulations: The interval over which we integrated, [a,b], was a finite interval, and the function f(x) was continuous on [a,b](ensuring that …

math.libretexts.org

Calculus II - Improper Integrals

In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. not infinite) value. Determining if they...

tutorial.math.lamar.edu

Then you can apply the same steps to your problem, and learn more than you would if we just gave you the complete solution.

Please write back to show us your work, so we can offer any additional help you might need. (I'm assuming your difficulty is not with the integration itself, by substitution or by formula, but with the infinite limit of integration.)

 

[mihalaras]

If you need an example of the steps (not for this problem, but one like it), see these textbooks, for example:

6.8: Improper Integration

When we defined the definite integral, we made two stipulations: The interval over which we integrated, [a,b], was a finite interval, and the function f(x) was continuous on [a,b](ensuring that …

math.libretexts.org

Calculus II - Improper Integrals

In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. not infinite) value. Determining if they...

tutorial.math.lamar.edu

Then you can apply the same steps to your problem, and learn more than you would if we just gave you the complete solution.

Please write back to show us your work, so we can offer any additional help you might need. (I'm assuming your difficulty is not with the integration itself, by substitution or by formula, but with the infinite limit of integration.)

This is exactly what I was looking for. I wanted to revise the method used for Improper Integrals. Thank you a lot!