Test the convergence of the following integral ∫∞−∞e−x1+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