Answer
Divergent
Work Step by Step
Let
\[I=\int_{0}^{\infty}\frac{1}{\sqrt[4]{1+x}}dx\;\;\;\ldots(1)\]
\[I=\lim_{t\rightarrow \infty}\int_{0}^{t}\frac{1}{\sqrt[4]{1+x}}dx\;\;\;\ldots(2)\]
\[I=\lim_{t\rightarrow \infty}\left[\frac{4}{3}(1+x^2)^{\frac{3}{4}}\right]_{0}^{t}\]
\[I=\lim_{t\rightarrow \infty}\left[\frac{4}{3}(1+t^2)^{\frac{3}{4}}-\frac{4}{3}\right]\]
\[I=\infty\]
Since limit on R.H.S. of (2) does not exist
So given improper integral (1) is divergent.
