ИНТЕГРАЛИ - моля за съдействие

[Гост]

[tex]\int[/tex]x2 In(x+1)dx

[Добромир Глухаров]

$\int x^2ln(x+1)dx=\int ln(x+1)d\frac{x^3}{3}=\frac{x^3}{3}ln(x+1)-\int\frac{x^3}{3}d(ln(x+1))=\frac{x^3ln(x+1)}{3}-\frac{1}{3}\int\frac{x^3+1-1}{x+1}dx=\\=\frac{x^3ln(x+1)}{3}-\frac{1}{3}\left(\int\frac{x^3+1}{x+1}dx-\int\frac{dx}{x+1}\right)=\frac{x^3ln(x+1)}{3}-\frac{1}{3}\left(\int(x^2-x+1)dx-ln(x+1)\right)=\\=\frac{x^3ln(x+1)}{3}-\frac{1}{3}\left(\frac{x^3}{3}-\frac{x^2}{2}+x-ln(x+1)\right)+C=\frac{x^3+1}{3}ln(x+1)-\frac{x}{18}(2x^2-9x+6)+C$