Question

$\int \frac{1}{\sqrt{8+3x-x^2}}dx$

Answer 1

$$\begin{gathered} \int \frac{dx}{\sqrt{8+3x-x^2}} \\ \Rightarrow \int \frac{dx}{\sqrt{8-\left(x^2-3x\right)}} \\ \Rightarrow \int \frac{dx}{\sqrt{8-\left(x^2-3x+{\left({\displaystyle \frac{3}{2}}\right)}^2-{\left({\displaystyle \frac{3}{2}}\right)}^2\right)}} \\ \Rightarrow \int \frac{dx}{\sqrt{8-{\left(x-{\displaystyle \frac{3}{2}}\right)}^2+{\displaystyle \frac{9}{4}}}} \\ \Rightarrow \int \frac{dx}{\sqrt{{\left({\displaystyle \frac{\sqrt{41}}{2}}\right)}^2-{\left(x-{\displaystyle \frac{3}{2}}\right)}^2}} \\ \Rightarrow {\sin }^{-1}\left(\frac{x-{\displaystyle \frac{3}{2}}}{{\displaystyle \frac{\sqrt{41}}{2}}}\right)+C \\ \Rightarrow {\sin }^{-1}\left(\frac{2x-3}{\sqrt{41}}\right)+C \end{gathered}$$