Answer
Diverges
Work Step by Step
An improper integral converges if the limit exists and is finite. The lower integration limit, 0, makes this integral improper. Replace 0 with variable $t$ and find the limit of the integral.
$\int_{0}^{1} \frac{1}{x^{5}}dx$ = $\lim\limits_{t \to 0} \int_{t}^{1}\frac{1}{x^{5}}dx$
Use the power rule $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ to integrate:
= $\lim\limits_{t \to 0} \frac{-1}{4x^{4}}\Big|_t^1$
Evaluate the limit:
= $\lim\limits_{t \to 0} \frac{-1}{4}+\frac{1}{4t^{4}}$
= $\frac{-1}{4} + \infty$
= $\infty$
The limit of this integral is not finite, so this integral diverges.
