Question

Пожалуйста решите интеграл

Answer 1

$\int \dfrac{2x-3}{\sqrt{-x^2-4x+5}}\, dx=\Big[\ -(x^2+4x-5)=-\Big((x+2)^2-4-5\Big)=9-(x+2)^2\ \Big]=\\\\\\=\int \dfrac{2x-3}{\sqrt{9-(x+2)^2}}\, dx=\Big[\ t=x+2\ ,\ x=t-2,\ dx=dt\ \Big]=\int \dfrac{2t-7}{\sqrt{9-t^2}}\, dt=\\\\\\=-\int \dfrac{-2t\, dt}{\sqrt{9-t^2}}-7\int \dfrac{dt}{\sqrt{9-t^2}}=-2\sqrt{9-t^2}-7\cdot arcsin\dfrac{t}{3}+C=\\\\\\=-2\sqrt{-x^2-4x+5}-7\, arcsin\dfrac{x+2}{3}+C$