1. $\displaystyle\int{\frac{1-x}{\sqrt{1-x-{{x}^{2}}}}}\,\mathrm{d}x$ 2. \(\displaystyle\int{\frac

Question

1. $\displaystyle\int{\frac{1-x}{\sqrt{1-x-{{x}^{2}}}}}\,\mathrm{d}x$ 2. $\displaystyle\int{\frac{1+2x}{\sqrt{3+2x-{{x}^{2}}}}}\,\mathrm{d}x$ 3. $\displaystyle\int{\frac{{{x}^{2}}-2x+6}{{{x}^{2}}+2x+6}}\,\mathrm{d}x$ 4. $\displaystyle\int{\frac{2-x}{4{{x}^{2}}+3x+1}}\,\mathrm{d}x$ 5. $\displaystyle\int{\frac{4x-5}{{{x}^{2}}+3x+2}}\,\mathrm{d}x$ 6. $\displaystyle\int{\frac{3x+7}{-3{{x}^{2}}+4x+6}}\,\mathrm{d}x$ 7. $\displaystyle\int{\frac{4{{x}^{2}}+2}{2{{x}^{2}}-x-1}}\,\mathrm{d}x$ 8. $\displaystyle\int{\frac{2x+3}{\sqrt{5-2x-{{x}^{2}}}}}\,\mathrm{d}x$ 9. $\displaystyle\int{\frac{{{x}^{2}}-x-11}{3-x-2{{x}^{2}}}}\,\mathrm{d}x$

Accepted Answer

1. $\frac{3}{2}{{\sin }^{-1}}\left( \frac{2x+1}{\sqrt{5}} ight)+\sqrt{1-x-{{x}^{2}}}+C$ 2. $3{{\sin }^{-1}}\left( \frac{x-1}{2} ight)-2\sqrt{3+2x-{{x}^{2}}}+C$ 3. $-2\ln \left| {{x}^{2}}+2x+6 ight|+\frac{4\sqrt{5}}{5}{{	an }^{-1}}\left( \frac{\sqrt{5}}{5}(x+1) ight)+x+C$ 4. $-\frac{1}{8}\ln \left| 4{{x}^{2}}+3x+1 ight|+\frac{19\sqrt{7}}{28}{{	an }^{-1}}\left( \frac{\sqrt{7}}{7}(8x+3) ight)+C$ 5. $\ln \left| \frac{{{\left| x+2 ight|}^{13}}}{{{\left| x+1 ight|}^{9}}} ight|+C$ 6. $\frac{9\sqrt{22}}{44}\ln \left| \frac{\sqrt{22}+3x-2}{\sqrt{22}-3x+2} ight|-\ln \left| \sqrt{6+4x-3{{x}^{2}}} ight|+C$ 7. $2x-\ln \left| 2x+1 ight|+2\ln \left| x-1 ight|+C$ 8. $-2\sqrt{5-2x-{{x}^{2}}}+{{\sin }^{-1}}\left( \frac{x+1}{\sqrt{6}} ight)+C$ 9. $-\frac{1}{2}x-\frac{11}{5}\ln \left| x-1 ight|+\frac{29}{20}\ln \left| 2x+3 ight|+C$

Techniques of Integration - Split Up Method

Techniques of Integration - Split Up Method