Question Number 20256 by tammi last updated on 24/Aug/17
$$\int\frac{{dx}}{\left({x}−\mathrm{4}\right)\sqrt{{x}+\mathrm{3}}} \\ $$
Answered by Tinkutara last updated on 24/Aug/17
$${x}\:+\:\mathrm{3}\:=\:{t}^{\mathrm{2}} \\ $$$${dx}\:=\:\mathrm{2}{tdt} \\ $$$${x}\:−\:\mathrm{4}\:=\:{t}^{\mathrm{2}} \:−\:\mathrm{7} \\ $$$$\int\frac{{dx}}{\left({x}\:−\:\mathrm{4}\right)\sqrt{{x}\:+\:\mathrm{3}}}\:=\:\int\frac{\mathrm{2}{tdt}}{\left({t}^{\mathrm{2}} \:−\:\mathrm{7}\right){t}}\:=\:\mathrm{2}\int\frac{{dt}}{{t}^{\mathrm{2}} \:−\:\mathrm{7}} \\ $$$$=\:\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{7}}}\right)\:\mathrm{log}\:\mid\frac{{t}\:−\:\sqrt{\mathrm{7}}}{{t}\:+\:\sqrt{\mathrm{7}}}\mid\:+\:{C} \\ $$$$=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}}}\:\mathrm{log}\:\mid\frac{\sqrt{{x}\:+\:\mathrm{3}}\:−\:\sqrt{\mathrm{7}}}{\:\sqrt{{x}\:+\:\mathrm{3}}\:+\:\sqrt{\mathrm{7}}}\mid\:+\:{C} \\ $$
Commented by tammi last updated on 25/Aug/17
$${thanks} \\ $$