Question

ИНТЕГРАЛЫ
Помогите, пожалуйста

Answer 1

$\int\limits { \frac{x^{3}}{ x^{2} -2} } \, dx = \frac{1}{2}* \int\limits { \frac{x^{2}}{ x^{2} -2} } \, d(x^{2}-2)=\frac{1}{2}* \int\limits { \frac{x^{2}-2+2}{ x^{2} -2} } \, d(x^{2}-2)= \\ =\frac{1}{2}* \int\limits {1 } \, d(x^{2}-2)+\frac{1}{2}* 2 \int\limits { \frac{1}{ x^{2} -2} } \, d(x^{2}-2)= \\ =\frac{1}{2}* ( x^{2} -2)+ln|x^{2}-2|+C$

Answer 2

 x³         |x²-2
-(x³-2x) |--------
-------    |  x
  2x                      x³/(x²-2) = x+(2x/(x²-2))

$\int \frac{x^3}{x^2-2}\, dx=\int (x+\frac{2x}{x^2-2})\, dx=\int x\, dx+\int \frac{2x\, dx}{x^2-2}=\\\\ =\frac{x^2}{2}+\int \frac{d(x^2-2)}{x^2-2}=\frac{x^2}{2}+ln|x^2-2|+C$