Question

Найти интеграл ∫(3х+5/х^2 -9x +1)dx

Answer 1

$\int \frac{3x+5}{x^2-9x+1}dx=\int \frac{3x+5}{(x-\frac{9}{2})^2-\frac{77}{4}}dx=[t=x-\frac{9}{2},x=t+\frac{9}{2},dx=dt]=\\\\=\int \frac{3t+\frac{37}{2}}{t^2-\frac{77}{4}}dt=\frac{3}{2}\int\frac{2t\, dt}{t^2-\frac{77}{4}}dt+\frac{37}{2}\int \frac{dt}{t^2-\frac{77}{4}}=\\\\=\frac{3}{2}\cdot ln|t^2-\frac{77}{4}|+\frac{37}{2}\cdot \frac{2}{2\cdot \sqrt{77}}\cdot ln|\frac{t-\frac{\sqrt{77}}{2}}{t+\frac{\sqrt{77}}{2}}|+C=$

$=\frac{3}{2}ln|x^2-9x+1|+\frac{37}{2\sqrt{77}}\cdot ln|\frac{2x-9-\sqrt{77}}{2x-9+\sqrt{77}}|+C$