Question

Нужна помощь


$intlimits^3_1 frac{dx}{ x^{2} + x^{3} }$

Answer 1

$int frac{1}{x^2+x^3}dx=int frac{dx}{x^2(1+x)};\\frac{1}{x^2(1+x)}=frac{A}{x^2}+frac{B}{x}+frac{C}{x+1}=frac{A(x+1)+Bx(x+1)+Cx^2}{x^2(x+1)}\\1=Ax+A+Bx^2+Bx+Cx^2\\x^2|B+C=0\\x; |A+B=0\\x^0|A=1\\B=-A=-1;; C=-B=1$

$int _1^3frac{dx}{x^2+x^3}=int _1^3frac{1}{x^2}dx-int _1^3frac{1}{x}dx+int _1^3frac{dx}{x+1}=\\=-frac{1}{x}|_1^3-ln|x||_1^3+ln|x+1||_1^3=-frac{1}{3}+1-ln3+ln1+ln4-ln2=\\=frac{2}{3}+lnfrac{4}{6}=frac{2}{3}+lnfrac{2}{3}$