Question

интеграл dx/(x-1)(X+3) помогите

Answer 1

$intlimits { frac{dx}{(x-1)(x+3)} } ,$
$frac{1}{(x-1)(x+3)}= frac{A}{x-1}+frac{B}{x+3}= frac{Ax+3A+Bx-B}{(x-1)(x+3)}= frac{(A+B)x+3A-B}{(x-1)(x+3)}$.
$left { {{A+B=0} atop {3A-B=1}} right.Rightarrow left { {{A= frac{1}{4} } atop {B=-frac{1}{4}}} right.$
$intlimits { frac{dx}{(x-1)(x+3)} } ,= frac{1}{4} intlimits { frac{dx}{x-1} } ,-frac{1}{4} intlimits { frac{dx}{x+3} } ,= frac{1}{4}ln|x-1|- frac{1}{4}ln|x+3|+C= frac{1}{4}ln frac{|x-1|}{|x+3|}+C$.