Answer
$$\frac{{900}}{{\ln 10}}$$
Work Step by Step
$$\eqalign{
& \int_{1/3}^{1/2} {\frac{{{{10}^{1/p}}}}{{{p^2}}}} dp \cr
& {\text{substitute }}u = \frac{1}{p},{\text{ }}du = - \frac{1}{{{p^2}}}dp \cr
& {\text{express the limits in terms of }}u \cr
& x = 1/2{\text{ implies }}u = 1/\left( {1/2} \right) = 2 \cr
& x = 1/3{\text{ implies }}u = 1/\left( {1/3} \right) = 3 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_{1/3}^{1/2} {\frac{{{{10}^{1/p}}}}{{{p^2}}}} dp = - \int_3^2 {{{10}^u}du} \cr
& {\text{by the formula }}\int {{a^u}} du = \frac{{{a^u}}}{{\ln a}} + C \cr
& {\text{letting }}a = 10 \cr
& = - \left. {\left( {\frac{{{{10}^u}}}{{\ln 10}}} \right)} \right|_3^2 \cr
& {\text{use the fundamental theorem}} \cr
& = - \frac{1}{{\ln 10}}\left( {{{10}^2} - {{10}^3}} \right) \cr
& {\text{simplify}} \cr
& = \frac{{900}}{{\ln 10}} \cr} $$
