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

In summary: Maybe he could elaborate on what he did wrong?In summary, the homework statement asks if an integral diverges or converges. The attempt at a solution provided an answer that it converges. However, the answer was considered to be incorrect and the student lost marks.
  • #1
Jeanclaud
16
0
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.
 
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  • #2
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 :sorry:
 
  • #3
Jeanclaud said:
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"?
 
  • #5
Mark44 said:
Was there any reason given why your answer was "all wrong"?
nope.
 

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  • #6
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 :woot:
 
  • #7
Jeanclaud said:
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.
 
  • #8
thanks you.
 
  • #9
Ray Vickson said:
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.
 
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  • #10
Mark44 said:
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.
 

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

1. What is the definition of integral convergence?

Integral convergence refers to the property of a mathematical series or integral where the value approaches a finite number as the terms or limits of integration increase. In other words, the series or integral "converges" to a specific value rather than diverging to infinity.

2. What does the function 1/(e^x sqrt(x)) represent?

The function 1/(e^x sqrt(x)) represents a type of exponential decay, where the value decreases rapidly as x increases. The square root in the denominator causes the function to approach 0 as x approaches infinity.

3. How is the convergence of this integral determined?

The convergence of this integral can be determined by evaluating the limit of the integral as x approaches infinity. If the limit is a finite number, then the integral converges. If the limit is infinity, then the integral diverges.

4. What is the significance of the "e" in the function?

The "e" in the function represents the mathematical constant, Euler's number, which is approximately equal to 2.71828. This constant is commonly used in exponential functions and has many applications in mathematics and science.

5. Can this integral be solved analytically?

Yes, this integral can be solved analytically using integration by parts or substitution. However, the resulting solution involves special functions such as the exponential integral or incomplete gamma function.

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