Question

Найти первообразную: 1) ∫ 3х²+2/ x²+2x dx ( dх -за дробью) 2) ∫ eˣdx/ eˣ +1

Answer 1

$1)\; \; \int \frac{3x^2+2}{x^2+2x}\, dx=\int \Big (3-\frac{6x-2}{x(x+2)}\Big )\, dx=3\int dx-\int \frac{dx}{x}+\int \frac{7\, dx}{x+2}=\\\\=3x-ln|x|+7ln|x+2|+C\; ;\\\\\star \; \; \frac{6x-2}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}\\\\x=0:\; \; A=\frac{-2}{2}=-1\\\\x=-2:\; B=\frac{-14}{-2}=7$

$2)\; \; \int \frac{e^{x}\, dx}{e^{x}+1}=[\, t=e^{x}\; ,\; x=lnt\; ,\; dx=\frac{dt}{t}\, ]=\int \frac{t\, dt}{t\cdot (t+1)}=\\\\=\int \frac{dt}{t+1}=ln|t+1|+C=ln|e^{x}+1|+C\; .$