Integral Convergence: Examining 1/(e^x sqrt(x))

[Jeanclaud]

http://‪C:\Users\johny\Downloads\q4.jpg 1. Homework Statement
Hi, so the question is I have to tell if this integral diverges or converges.(without solving it)
integral(1/(e^x sqrt(x)))dx from 1 to +inf

Homework Equations

integration techniques.

The Attempt at a Solution

my answer: let 1/e^x >1/(e^x sqrt(x))
then I solved the definite integral(1/e^x)from 1 to +inf and got 1/e which means it converges.
so the given integral has to converge also since it is smaller than the 1/e^x.
that was my answer in the exam but they considered it all wrong so please can anybody tell me the reason. Thank you.

 

[Simon Phoenix]

Looking at the form of this integral $$ \int_1^\infty \frac 1 { \sqrt x e^x} \, dx $$

it seems fairly clear we're expecting convergence here - so for the comparison test we are looking for a larger integral on the domain that converges. We have
$$ \frac 1 { \sqrt x e^x } \leq \frac 1 { e^x } $$ on this domain. Everything's all positive so we're good to go.

The integral $$ \int_1^\infty \frac 1 { e^x } \, dx $$ is clearly convergent, so our integral of interest is also convergent.

I can't see why your answer was considered to be incorrect either

 

[Mark44]

so the given integral has to converge also since it is smaller than the 1/e^x.
that was my answer in the exam but they considered it all wrong

Was there any reason given why your answer was "all wrong"?

 

[Mark44]

BTW, the link in your post isn't valid URL: http://‪C:\Users\johny\Downloads\q4.jpg
You can use the UPLOAD button in the lower right corner to actually upload your image.

 

[Jeanclaud]

Was there any reason given why your answer was "all wrong"?

nope.

 

[Simon Phoenix]

Well, I can't see why your answer was considered to be wrong. Just a silly thought; you did actually copy the integral down from the exam paper correctly?

That's the only thing I can think of because your answer looks OK to me.

You could probably structure your answer a bit better - does your answer look anything like the worked examples in your textbooks in terms of how it's laid out? Try to copy this 'style' and set out your answers in a clear step-by-step fashion and explain what you're doing (only takes a few words here and there). It's a bit of a pain to do this initially but it becomes easier with practice and eventually becomes second nature - and you (should) find your understanding and maybe even your marks improve the more you do this

 

[Ray Vickson]

nope.

I second the remarks of Simon Phoenix regarding your presentation style. I would add that some of the things you wrote are technically wrong without further qualification. In particular, the inequality $e^{-x} > e^{-x}/\sqrt{x}$ is false when $x < 1$, but true when $x > 1$. You could say something like "since we want x > 1, ..." and then what you wrote would be correct. Just a few words of explanation is all you need; it would take < 5 seconds to write them.

 

[Jeanclaud]

thanks you.

 

[Mark44]

second the remarks of Simon Phoenix regarding your presentation style. I would add that some of the things you wrote are technically wrong without further qualification. In particular, the inequality e^−x > e^−x/ √x is false when x < 1, but true when x > 1.

But since the interval of integration is $[1, \infty)$, I don't think it's absolutely necessary to stipulate that x > 1.

 

[Ray Vickson]

But since the interval of integration is $[1, \infty)$, I don't think it's absolutely necessary to stipulate that x > 1.

We are all just trying to figure out why he lost marks on correct work.