Question

интеграл dx/sgrt (-9x^2 +5x+1)​

Answer 1

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