Answer
Convergent$\;\;,\;\;$ $\large\frac{3}{2}(5)^{\frac{2}{3}}$
Work Step by Step
Let \[I=\int_{0}^{5}\frac{1}{\sqrt[3]{5-x}}dx\]
Since 5 is point of infinite discontinuity of integrand $\frac{1}{\sqrt[3]{5-x}}$
\[I= \int_{0}^{5}\frac{1}{\sqrt[3]{5-x}}dx= \lim_{t\rightarrow 5^-}\int_{0}^{t}\frac{1}{\sqrt[3]{5-x}}dx\]
\[I=\lim_{t\rightarrow 5^-}\int_{0}^{t}\frac{1}{\sqrt[3]{5-x}}dx\;\;\;\ldots(1)\]
\[I=\lim_{t\rightarrow 5^-}\int_{0}^{t}(5-x)^{\frac{-1}{3}}dx\]
\[I=\lim_{t\rightarrow 5^-}\left[-\frac{3}{2}(5-x)^{\frac{2}{3}}\right]_{0}^{t}\]
\[I=\lim_{t\rightarrow 5^-}\left[-\frac{3}{2}(5-t)^{\frac{2}{3}}+\frac{3}{2}(5)^{\frac{2}{3}}\right]\]
\[I=\frac{3}{2}(5)^{\frac{2}{3}}\]
Since limit on R.H.S. of (1) exists so $I$ is convergent and $I=\frac{3}{2}(5)^{\frac{2}{3}}$.
