Question

найти интеграл
∫dx/ корень 2x^2-x+3

Answer 1

$int frac{dx}{2x^2-x+3}=Big [; 2x^2-x+3=2(x^2-frac{1}{2}x+frac{3}{2})=2cdot ((x-frac{1}{4})^2-frac{1}{16}+frac{3}{2})=\\=2cdot ((x-frac{1}{4})^2+frac{23}{8}); Big ]=frac{1}{2}int frac{dx}{(x-frac{1}{4})^2+frac{23}{8}}=frac{1}{2}int frac{d(x-frac{1}{4})}{(x-frac{1}{4})^2+frac{23}{8}}=\\=frac{1}{2}cdot frac{2sqrt2}{sqrt{23}}cdot arctgfrac{2sqrt2, (x-frac{1}{4})}{sqrt{23}}+C=sqrt{frac{2}{23}}cdot arctgfrac{sqrt2, (4x-1)}{2sqrt{23}}+C$