Answer
$${\text{The integral diverges}}$$
Work Step by Step
$$\eqalign{
& \int_3^4 {\frac{{dz}}{{{{\left( {z - 3} \right)}^{3/2}}}}} \cr
& {\text{The integrand is not defined for }}x = 3,{\text{ then}} \cr
& \int_3^4 {\frac{{dz}}{{{{\left( {z - 3} \right)}^{3/2}}}}} = \mathop {\lim }\limits_{a \to {3^ + }} \int_a^4 {\frac{{dz}}{{{{\left( {z - 3} \right)}^{3/2}}}}} \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{a \to {3^ + }} \left[ {\frac{{{{\left( {z - 3} \right)}^{ - 1/2}}}}{{ - 1/2}}} \right]_a^4 \cr
& = - 2\mathop {\lim }\limits_{a \to {3^ + }} \left[ {\frac{1}{{\sqrt {z - 3} }}} \right]_a^4 \cr
& = - 2\mathop {\lim }\limits_{a \to {3^ + }} \left[ {\frac{1}{{\sqrt {4 - 3} }} - \frac{1}{{\sqrt {a - 3} }}} \right] \cr
& = - 2\mathop {\lim }\limits_{a \to {3^ + }} \left[ {1 - \frac{1}{{\sqrt {a - 3} }}} \right] \cr
& {\text{Evaluating the limit}} \cr
& = \infty \cr
& {\text{The integral diverges}} \cr} $$
