Question Number 20238 by tammi last updated on 24/Aug/17
$$\int\frac{{dx}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}} \\ $$
Commented by tammi last updated on 24/Aug/17
$${this}\:{answer}\:{is}\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{c} \\ $$$${i}\:{know}\:{the}\:{answer}\:{but}\:{can}\:{not}\:{solve}\:{this}\:{prblm}..{help} \\ $$
Answered by $@ty@m last updated on 25/Aug/17
$$=\int\frac{{dx}}{{x}^{\mathrm{2}} −\mathrm{2}.\frac{\mathrm{1}}{\mathrm{2}}{x}+\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{1}} \\ $$$$=\int\frac{{dx}}{\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}{tan}^{−\mathrm{1}} \frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}}{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}{tan}^{−\mathrm{1}} \frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}+{C} \\ $$