Let \[ I = \int {\frac{{dt}}{{\sqrt {3t - 2{t^2}} }}} = \frac{1}{{\sqrt 2 }}\int {\frac{{dt}}{{\sqrt { - \left( {{t^2} - \frac{3}{2}t} \right)} }}} \]


\[ = \frac{1}{{\sqrt 2 }}\int {\frac{{dt}}{{\sqrt { - \left[ {\left( {{t^2} - 2 \cdot \frac{1}{2} \cdot \frac{3}{2}t} \right) + {{\left( {\frac{3}{4}} \right)}^2} - {{\left( {\frac{3}{4}} \right)}^2}} \right]} }}} \]

\[ = \frac{1}{{\sqrt 2 \int {\frac{{dt}}{{\sqrt { - \left[ {{{\left( {t - \frac{3}{4}} \right)}^2} - {{\left( {\frac{3}{4}} \right)}^2}} \right]} }}} }}\]

\[ = \frac{1}{{\sqrt 2 \int {\frac{{dt}}{{\sqrt {{{\left( {\frac{3}{4}} \right)}^2} - {{\left( {t - \frac{3}{4}} \right)}^2}} }}} }}\]

\[ = \frac{1}{{\sqrt 2 }}{\sin ^{ - 1}}\left( {\frac{{t - \frac{3}{4}}}{{\frac{3}{4}}}} \right) + C = \frac{1}{{\sqrt 2 }}{\sin ^{ - 1}}\left( {\frac{{4t - 3}}{3}} \right) + C\]

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