Results 1 to 3 of 3

Math Help - change of variable question (changing of signs)

  1. #1
    Senior Member
    Joined
    Nov 2010
    From
    Hong Kong
    Posts
    255

    change of variable question (changing of signs)

    Question (Schaum Advanced Calculus). Test for convergence:

     \int^{-1}_{-{infty}} \frac{\exp{x}}{x}\;{dx}

    My thinking. Substitute y=-x and then deal with a simpler construction  \frac{\exp{(-y)}}{y}. This is what the model answer is saying, too: -  \int^{infty}_{1}} \frac{\exp{(-y)}}{y}\;{dy} converges by comparison to the similar integral of \exp{(-y)}


    My problem is to understand why the new integral is negative. Here is my list of changes of signs during the substitution y=-x:

    1) x in the integrand changes to -y, so there is the first new '-' sign

    2) dy/dx is -1 (differentiating y=-x gives a '-')

    3) since the limits of integration are changing from - \infty to \infty and from -1 to 1, we need to swap them (put the infinity to the top, and the 1 to the bottom), resulting in yet another '-' sign - number 3.

    Therefore, total minuses: 3; overall, the integral becomes negative. Is that right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by Volga View Post
    Is that right?
    Yes:

    \displaystyle\int_{-\infty}^{-1}\dfrac{e^x}{x}dx=\displaystyle\int_{+\infty}^{1}  \dfrac{e^{-y}}{-y}(-dy)=\displaystyle\int_{+\infty}^{1}\dfrac{e^{-y}}{y}dy=-\displaystyle\int_{1}^{+\infty}\dfrac{e^{-y}}{y}dy

    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2010
    From
    Hong Kong
    Posts
    255
    Thank you! I am happy now )))
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Change of variable triple integral question.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 16th 2011, 12:07 AM
  2. help with a question about change of variable
    Posted in the Statistics Forum
    Replies: 5
    Last Post: October 16th 2010, 10:06 PM
  3. change in variable question
    Posted in the Statistics Forum
    Replies: 7
    Last Post: October 16th 2010, 06:06 PM
  4. changing variable
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 20th 2009, 04:12 PM
  5. multintegral question with change of variable
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 30th 2009, 03:57 PM

Search Tags


/mathhelpforum @mathhelpforum