Question

Determine whether the integral converges or diverges. Find the value of the integral if it converges.

Answer 1

\[\int_0^1x^{-4/3}~dx\]
The integrand is undefined at \(x=0\), so you have an improper integral. Rewrite it as a limit:
\[\lim_{a\to0}\int_a^1x^{-4/3}~dx\]
Power rule and FTC give you
\[\lim_{a\to0}\left[\frac{x^{-1/3}}{-\frac{1}{3}}\right]_a^1\\
-3~\lim_{a\to0}\left[x^{-1/3}\right]_a^1\\
-3~\lim_{a\to0}\left[1^{-1/3}-a^{-1/3}\right]\\
-3\left(\frac{1}{\sqrt[3]1}-\lim_{a\to0}\frac{1}{\sqrt[3]a}\right)\\
\]

Answer 2

So, the integral converges if the limit exists and is finite; diverges otherwise.