Integration of (5x+4)÷(x²+3x+2)^1/2

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{I}=\int \frac{\left(5\mathrm{x}+4\right)}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ \mathrm{Try} \mathrm{to} \mathrm{create} \mathrm{derivative} \mathrm{of} \left({\mathrm{x}}^2+3\mathrm{x}+2\right) \mathrm{i}.\mathrm{e} \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^2+3\mathrm{x}+2\right)=2\mathrm{x}+3, \mathrm{in} \mathrm{the} \\ \mathrm{numerator}. \\ \int \frac{\left(5\mathrm{x}+4\right)}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ =5\int \frac{\mathrm{x}+{\displaystyle \frac{4}{5}}}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ =\frac{5}{2}\int \frac{2\mathrm{x}+{\displaystyle \frac{8}{5}}}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \left[\mathrm{Multiply} \mathrm{and} \mathrm{divide} \mathrm{by} 2\right] \\ =\frac{5}{2}\int \frac{2\mathrm{x}+3+{\displaystyle \frac{8}{5}}-3}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ =\frac{5}{2}\int \frac{2\mathrm{x}+3-{\displaystyle \frac{7}{5}}}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ \Rightarrow \mathrm{I}=\frac{5}{2}\left[\int \frac{2\mathrm{x}+3}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx}-\frac{7}{5}\int \frac{1}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx}\right] ;\left(\mathrm{i}\right) \\ \mathrm{Let} \mathrm{J}=\int \frac{2\mathrm{x}+3}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ \mathrm{Set} {\mathrm{x}}^2+3\mathrm{x}+2=\mathrm{t} \\ \Rightarrow \left(2\mathrm{x}+3\right).\mathrm{dx}=\mathrm{dt} \\ \Rightarrow \mathrm{J}=\int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}}}=2\sqrt{\mathrm{t}}=2\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2} \\ \mathrm{Let} \mathrm{K}=\int \frac{1}{\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}}.\mathrm{dx} \\ =\int \frac{1}{\sqrt{{\left(\mathrm{x}+{\displaystyle \frac{3}{2}}\right)}^2-{\displaystyle \frac{9}{4}}+2}}.\mathrm{dx} \\ =\int \frac{1}{\sqrt{{\left(\mathrm{x}+{\displaystyle \frac{3}{2}}\right)}^2-{\displaystyle \frac{1}{4}}}}.\mathrm{dx} \\ =\int \frac{1}{\sqrt{{\left(\mathrm{x}+{\displaystyle \frac{3}{2}}\right)}^2-{{\displaystyle \left(\frac{1}{2}\right)}}^2}}.\mathrm{dx} \\ =\ln \left|\mathrm{x}+\frac{3}{2}-\sqrt{{\left(\mathrm{x}+\frac{3}{2}\right)}^2{-}^2\left(\frac{1}{2}\right)}\right| \\ \Rightarrow \mathrm{K}=\ln \left|\mathrm{x}+\frac{3}{2}-\sqrt{{\mathrm{x}}^2+3\mathrm{x}+2}\right| \\ \mathrm{Put} \mathrm{value} \mathrm{of} \mathrm{J} \mathrm{and} \mathrm{K} \mathrm{in} \left(\mathrm{i}\right) \\ \Rightarrow \mathbf{I}\mathbf{=}\frac{\mathbf{5}}{\mathbf{2}}\left[\mathbf{2}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}\mathbf{-}\frac{\mathbf{7}}{\mathbf{5}}\mathbf{ln}\left|\mathbf{x}\mathbf{+}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{-}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}}\right|\right]\mathbf{+}\mathbf{C} \end{gathered}$$
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