Answer
$$\frac{1}{2}\left[ {{{\ln }^2}\left( 2 \right) - {{\ln }^2}\left( {\frac{3}{2}} \right)} \right]$$
Work Step by Step
$$\eqalign{
& \int_{5/12}^{3/4} {\frac{{{{\sinh }^{ - 1}}x}}{{\sqrt {{x^2} + 1} }}} dx \cr
& {\text{set }}u = {\sinh ^{ - 1}}x{\text{ then }}du = \frac{1}{{\sqrt {{x^2} + 1} }}dx \cr
& {\text{switch the limits of integration}} \cr
& u = {\sinh ^{ - 1}}x,\,\,\,\,x = 5/12 \to {\sinh ^{ - 1}}\left( {5/12} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 3/4 \to {\sinh ^{ - 1}}\left( {3/4} \right) \cr
& {\text{use the change of variable}} \cr
& \int_{5/12}^{3/4} {\frac{{{{\sinh }^{ - 1}}x}}{{\sqrt {{x^2} + 1} }}} dx = \int_{{{\sinh }^{ - 1}}\left( {5/12} \right)}^{{{\sinh }^{ - 1}}\left( {3/4} \right)} u du \cr
& {\text{integrate and evaluate the limits}} \cr
& = \frac{{{{\left( {{{\sinh }^{ - 1}}\left( {3/4} \right)} \right)}^2}}}{2} - \frac{{{{\left( {{{\sinh }^{ - 1}}\left( {5/12} \right)} \right)}^2}}}{2} \cr
& = \frac{1}{2}\left[ {{{\sinh }^{ - 1}}\left( {\frac{3}{4}} \right) - {{\sinh }^{ - 1}}\left( {\frac{5}{{12}}} \right)} \right] \cr
& {\text{Using the theorem 6}}{\text{.10 to express the answer in terms of logarithms}}: \cr
& {\text{ }}{\sinh ^{ - 1}}x = \ln \left( {x + \sqrt {{x^2} + 1} } \right) \cr
& \cr
& = \frac{1}{2}\left[ {{{\ln }^2}\left( {\frac{3}{4} + \sqrt {{{\left( {\frac{3}{4}} \right)}^2} + 1} } \right) - {{\ln }^2}\left( {\frac{5}{{12}} + \sqrt {{{\left( {\frac{5}{{12}}} \right)}^2} + 1} } \right)} \right] \cr
& {\text{Simplify}} \cr
& = \frac{1}{2}\left[ {{{\ln }^2}\left( {\frac{3}{4} + \frac{5}{4}} \right) - {{\ln }^2}\left( {\frac{5}{{12}} + \frac{{13}}{{12}}} \right)} \right] \cr
& = \frac{1}{2}\left[ {{{\ln }^2}\left( 2 \right) - {{\ln }^2}\left( {\frac{3}{2}} \right)} \right] \cr} $$
