SOLUTION: Please help me solve this equation:Use the Comparison Theorem to determine whether the following integrals are convergent or divergent. a. {{{ int ( root(3,x)/(sqrt(x)-pi), dx

Algebra ->  Test -> SOLUTION: Please help me solve this equation:Use the Comparison Theorem to determine whether the following integrals are convergent or divergent. a. {{{ int ( root(3,x)/(sqrt(x)-pi), dx      Log On


   

Question 363011: Please help me solve this equation:Use the Comparison Theorem to determine whether the following integrals are
convergent or divergent.
a. +int+%28+root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%2C+dx%2C+1%2C+infinity+%29+
b. +int+%28+e%5E%28-x%5E2%29%2C+dx%2C+1%2C+infinity+%29+
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
a. Let w+=+sqrt%28x%29-pi. Then x+=+%28w%2Bpi%29%5E2, and dx+=+2sqrt%28x%29dw. Then+%28root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%29dx++=+%28%28w%2Bpi%29%5E%285%2F3%29%2Fw%29dw. Now
%28w%2Bpi%29%5E%285%2F3%29%2Fw+%3E+=+w%5E%285%2F3%29%2Fw++=++w%5E%282%2F3%29. It is enough to check the integral for x%3E=pi%5E2 for divergence/convergence (Why?). We have then
But the last integral has infinite value, and thus, the given integral itself has infinite value. The integral is divergent.
b.. Therefore the given integral is convergent.