Monday, April 23, 2018

real analysis - Test the convergence of the integral intinftyinftyfracex1+x2.



Test the convergence of the following integral ex1+x2.I can not find the indefinite integral of the integrand so that we can check at the limits and Also I can not apply any theorem about convergence , like Ables test, Dirichlet's test...etc... Can anyone help me?


Answer



Follow @Solitary comment. Let f(x) be the integrand function.
Notice that
lim
hence there is some b<0 such that for all $x1 hence for all a \int_{a}^{b} f(x) \ge b-a \to +\infty
as a\to -\infty.



Hence
\int_{-\infty}^0 f(x) = +\infty.
This assumes that you are speaking of improper integrals. If you are speaking of Lebesgue integrals the solution is even simpler...


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...