Question

Очень сильно нужна помощь. Даю 50.

Answer 1

$1)\; \; \int \frac{2x-1}{(x-1)(x-2)}dx=-\int \frac{dx}{x-1}+3\cdot \int \frac{dx}{x-2} =\\\\=-ln|x-1|+3ln|x-2|+C\\\\\frac{2x-1}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}\\x=1:\; \; A=\frac{2-1}{1-2}=-1\\x=2:\; \; B=\frac{4-1}{2-1}=3$

$2)\; \; \int \frac{3x^2+2x-3}{x^3-x}=\int \frac{3x^2+2x-3}{x(x-1)(x+1)}=3\int \frac{dx}{x}+\int \frac{dx}{x-1}-\int \frac{dx}{x+1}=\\\\=3ln|x|+ln|x-1|-ln|x+1|+C=ln\Big |\frac{x^3(x-1)}{x+1}\Big |+C\\\\\frac{3x^2+2x-3}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}\\x=0:\; \; A=\frac{-3}{(0-1)(0+1)}=3\\x=1:\; \; B=\frac{3+2-3}{1\cdot (1+1)}=1\\x=-1:\; \; C=\frac{3-2-3}{-1\cdot (-1-1)}=-1$

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