Answer
$$1$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\frac{1}{{x\sqrt x }}} dx \cr
& {\text{multiply}} \cr
& = \int_1^4 {\frac{1}{{{x^{3/2}}}}} dx \cr
& {\text{negative exponent}} \cr
& = \int_1^4 {{x^{ - 3/2}}} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \left( {\frac{{{x^{ - 1/2}}}}{{ - 1/2}}} \right)_1^4 \cr
& = - 2\left( {\frac{1}{{\sqrt x }}} \right)_1^4 \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = - 2\left( {\frac{1}{{\sqrt 4 }} - \frac{1}{{\sqrt 1 }}} \right) \cr
& = - 2\left( {\frac{1}{2} - \frac{1}{1}} \right) \cr
& = - 2\left( { - \frac{1}{2}} \right) \cr
& = 1 \cr} $$
