Question

Determine whether the improper integral converges or diverges, and find the value of each that converges.

∫^[infinity]_0 dx/(4x+1)^3

Answer 1

It converges to 1/8.

Explanation:

We are given the integral:
`\int\limits^\infty_0 (dx)/((4x+1)^3)`

`\int\limits^\infty_0 (dx)/((4x+1)^3)= \lim_(t \to \infty) \int\limits^t_0 (dx)/((4x+1)^3)=\lim_(t \to \infty)  (-1)/(8(4x+1)^2)|^t_0 = 0-(-(1)/(8))=(1)/(8)`

So it is convergent and converges to 1/8.