Mathematics

Solve $$\displaystyle\int {\dfrac{1}{{\sqrt {8 + 3x - {x^2}} }}} \,dx\,$$


SOLUTION
$$\begin{array}{l}\displaystyle\int {\dfrac{{dx}}{{\sqrt {8 + 3x - {x^2}} }}} \\ = \displaystyle\int {\dfrac{{dx}}{{ - \sqrt {{x^2} - 3x - 8} }}} \\ = \displaystyle\int {\dfrac{{dx}}{{ - \sqrt {\left( {{x^2} - 2.\dfrac{3}{2}.x + \dfrac{9}{4}} \right) - \dfrac{9}{4} - 8} }}} \\ = \displaystyle\int {\dfrac{{dx}}{{ - \sqrt {{{\left( {x - \dfrac{3}{2}} \right)}^2} - \dfrac{{41}}{4}} }}} \\ = \displaystyle\int {\dfrac{{dx}}{{\sqrt {\dfrac{{41}}{4} - {{\left( {x - \dfrac{3}{2}} \right)}^2}} }}} \\ = \displaystyle\int {\dfrac{{dx}}{{\sqrt {{{\left( {\dfrac{{\sqrt {41} }}{2}} \right)}^2} - {{\left( {x - \dfrac{3}{2}} \right)}^2}} }}} \\ = {\sin ^{ - 1}}\dfrac{{x - \dfrac{3}{2}}}{{\dfrac{{\sqrt {41} }}{2}}} + C\\ = {\sin ^{ - 1}}\left( {\dfrac{{2x - 3}}{{\sqrt {41} }}} \right) + C\end{array}$$
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Subjective Medium Published on 17th 09, 2020
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