Question

$\int \frac{x^2}{\left(x-1\right)\sqrt{x+2}}dx$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ I=\int \frac{x^2}{\left(x-1\right)\sqrt{x+2}}dx \\ \mathrm{Putting}x+2=t^2 \\ x=t^2-2 \\ \mathrm{Diff}\mathrm{both}\mathrm{sides} \\ dx=2tdt \\ I=\int \frac{{\left(t^2-2\right)}^2}{\left(t^2-2-1\right)t}2tdt \\ =2\int \frac{{\left(t^2-2\right)}^{2}dt}{t^2-3} \\ =2\int \frac{\left(t^4-4t^2+4\right)}{t^2-3}dt \\ \mathrm{Dividing}\mathrm{numerator}\mathrm{by}\mathrm{denominator},\mathrm{we}\mathrm{get} \\ {\mathrm{t}}^2-1 \\ {\mathrm{t}}^2-3\overline{){\mathrm{t}}^4-4{\mathrm{t}}^2+4} \\ {\mathrm{t}}^4-3{\mathrm{t}}^2 \\ -+ \\ \overline{-{\mathrm{t}}^2+4} \\ -{\mathrm{t}}^2+3 \\ +- \\ \overline{1} \\ \therefore I=2\int \left(t^2-1+\frac{1}{t^2-3}\right)dt \\ =2\int t^{2}dt-2\int dt+2\int \frac{dt}{t^2-{\left(\sqrt{3}\right)}^2} \\ =2\left[\frac{t^3}{3}\right]-2t+2\times \frac{1}{2\sqrt{3}}\log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+\mathrm{C} \\ =\frac{2}{3}{\left(\sqrt{x+2}\right)}^3-2\sqrt{x+2}+\frac{1}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+\mathrm{C} \\ =\frac{2}{3}{\left(x+2\right)}^{\frac{3}{2}}-2\sqrt{x+2}+\frac{1}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+\mathrm{C} \end{gathered}$$