change of variable question (changing of signs)
Question (Schaum Advanced Calculus). Test for convergence:
My thinking. Substitute y=-x and then deal with a simpler construction . This is what the model answer is saying, too: - converges by comparison to the similar integral of
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 - to 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?