Question

интеграл dx/sgrt (-9x^2 +5x+1)​

Answer 1

$int frac{dx}{-9x^2+5x+1}=int frac{dx}{-9(x^2-frac{5}{9}+frac{1}{9})}=-frac{1}{9}int frac{dx}{(x-frac{5}{18})^2-(frac{5}{18})^2+frac{1}{9}}=\\=-frac{1}{9}int frac{dx}{(x-frac{5}{18})^2+frac{11}{324}}=-frac{1}{9}int frac{d(x-frac{5}{18})}{(x-frac{5}{18})^2+(frac{sqrt{11}}{18})^2}=\\=-frac{1}{9}cdot frac{18}{sqrt{11}}cdot arctgfrac{18cdot (x-frac{5}{18})}{sqrt{11}}+C=-frac{2}{sqrt{11}}cdot arctgfrac{18x-5}{sqrt{11}}+C; ;\\\P.S.; ; ; x^2pm px+q=Big (xpm frac{p}{2}Big )^2-Big (frac{p}{2}Big )^2+q$