Question

Найти интеграл dx/(2x^2-11x+2)

Answer 1

$intfrac{1}{2x^2-11x+2} dx = frac{1}{2}intfrac{1}{x^2-frac{11}{2}x+1} dx = frac{1}{2}intfrac{1}{(x-frac{11}{4})^2 - frac{121}{16}+1} dx =\\\ frac{1}{2}intfrac{1}{(x-frac{11}{4})^2 - frac{121}{16}+1} dx = frac{1}{2}intfrac{1}{(x-frac{11}{4})^2 - (frac{sqrt{105}}{4})^2} dx =$

 

 

$frac{1}{2}intfrac{1}{(x-frac{11}{4} - frac{sqrt{105}}{4})(x-frac{11}{4} + frac{sqrt{105}}{4})} dx = frac{1}{2}intfrac{1}{(x - frac{sqrt{105} + 11}{4})(x + frac{sqrt{105} - 11}{4})} dx=$

 

 

$frac{1}{2}intfrac{frac{2}{sqrt{105}}}{x - frac{sqrt{105} + 11}{4}} - frac{1}{2}intfrac{frac{2}{sqrt{105}}}{x + frac{sqrt{105} - 11}{4}} dx =\\\ frac{1}{sqrt{105}}intfrac{1}{x - frac{sqrt{105} + 11}{4}} dx - frac{1}{sqrt{105}}intfrac{1}{x + frac{sqrt{105} - 11}{4}} dx =$

 

 

$frac{1}{sqrt{105}}ln|x - frac{sqrt{105} + 11}{4}| - frac{1}{sqrt{105}}ln|x + frac{sqrt{105} - 11}{4}| + C =$

 

 

$frac{1}{sqrt{105}}ln|frac{x - frac{sqrt{105} + 11}{4}}{x + frac{sqrt{105} - 11}{4}}| + C =frac{1}{sqrt{105}}ln|frac{4x - sqrt{105} - 11}{4x + sqrt{105} - 11}| + C$