Using the Comparison Theorem to determine if this integral is convergent or divergent.
[Int] from 0-->infinity [ x / ( x^3 + 1) ] dx
First i compare x^3 + 1 > x^3, thus x/(x^3+1) < x/x^3 = 1/x^2
But the integral of 1/x^2 can only be evaluated from 1 --> infinity. That integral is finite so it converges.
Thus the integral from 1--> infinity of [ x / (x^3 + 1)] dx converges. But the question asks for 0--> infinity.
So how do i know if the integral from 0--> 1 of [ x/ ( x^3 + 1 ) ] dx converges or not?
[Int] from 0-->infinity [ x / ( x^3 + 1) ] dx
First i compare x^3 + 1 > x^3, thus x/(x^3+1) < x/x^3 = 1/x^2
But the integral of 1/x^2 can only be evaluated from 1 --> infinity. That integral is finite so it converges.
Thus the integral from 1--> infinity of [ x / (x^3 + 1)] dx converges. But the question asks for 0--> infinity.
So how do i know if the integral from 0--> 1 of [ x/ ( x^3 + 1 ) ] dx converges or not?