TITLE: Test for convergence for improper integral $1/x^x$
QUESTION [5 upvotes]: I am having trouble determining if this is convergence or divergence
$$\int^1_0 1/x^x dx$$

REPLY [2 votes]: There is another way which is based on Caparison Test again. In fact, the following limit says that the improper integral is convergent: $$\lim_{x\to 0^+}~~x^{1/2}\times\frac{1}{x^x}=0<\infty$$