For what values of M does this integral converge?

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In summary, the integral \int_{-\infty}^{0}\frac{x.e^x}{(1-e^{2x})^M}dx converges for values of M less than 2, and the limit as x approaches 0 is equal to (-1)^M/2.
  • #1
nonaa
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For what values of M does this integral converge?

[tex]\int_{-\infty}^{0}\frac{x.e^x}{(1-e^{2x})^M}dx[/tex]


My pathetic attempt for solution ended here:

[tex]\frac{2x.e^x}{2.(-1)^M(e^{2x}-1)^M}.x^{M-1}\stackrel{x\rightarrow0}{\longrightarrow}\frac{(-1)^M}{2}[/tex]

[tex]M-1<1, M<2[/tex]

Am I going in the right direction? And what to do if [tex]x\rightarrow\infty[/tex]?
 
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  • #2
Welcome to PF!

Hi nonaa! Welcome to PF! :smile:

No, you're misreading the question.

It means for what values of M is the whole integral finite. :wink:
 
  • #3


Hi, I'm here for almost a day but you people are so nice and helpful :) I'm glad I've found this forum :)

And how to do this? I've never solved such type of problems before. Would you give me a little hint?
 

FAQ: For what values of M does this integral converge?

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used in calculus to calculate the total value of a function over a given interval.

How do you determine if an integral converges?

An integral converges if its value approaches a finite number as the interval approaches infinity. This can be determined by evaluating the limit of the integral as the interval approaches infinity.

What is the significance of the value of M in determining the convergence of an integral?

The value of M represents the upper limit of the interval over which the integral is being evaluated. It is used to determine if the integral converges or diverges.

What are some common methods for evaluating the convergence of an integral?

Some common methods include the comparison test, the ratio test, and the integral test. These methods involve comparing the given integral to a known converging or diverging series.

Are there any special cases where an integral may converge for certain values of M?

Yes, there are some special cases where an integral may converge for certain values of M. For example, if the integral is of a continuous and positive function, it will always converge regardless of the value of M.

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