Answer
$$\frac{{180}}{{\ln 10}}$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\frac{{{{10}^{\sqrt x }}}}{{\sqrt x }}} \cr
& {\text{substitute }}u = \sqrt x ,{\text{ }}du = \frac{1}{{2\sqrt x }}dx \cr
& {\text{express the limits in terms of }}u \cr
& x = \pi /2{\text{ implies }}u = \sqrt 4 = 2 \cr
& x = 0{\text{ implies }}u = \sqrt 1 = 1 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_1^4 {\frac{{{{10}^{\sqrt x }}}}{{\sqrt x }}} = 2\int_1^2 {{{10}^u}} du \cr
& {\text{by the formula }}\int {{a^u}} du = \frac{{{a^u}}}{{\ln a}} + C \cr
& {\text{letting }}a = 10 \cr
& = 2\left. {\left( {\frac{{{{10}^u}}}{{\ln 10}}} \right)} \right|_1^2 \cr
& {\text{use the fundamental theorem}} \cr
& = \frac{2}{{\ln 10}}\left( {{{10}^2} - {{10}^1}} \right) \cr
& = \frac{{180}}{{\ln 10}} \cr} $$
