Question

интеграл..............................................

Answer 1

$\int \frac{(x-1) \cdot (x+1)}{(x-2) \cdot (x-1) \cdot (x+2)} \, dx= \int \frac{ x+1}{(x-2)\cdot (x+2)} \, dx = (*) \\ \\ \frac{A}{x-2} + \frac{B}{x+2}=\frac{x+1}{(x-2)\cdot (x+2)} \\ \\ A(x+2) + B(x-2)=x+1 \\ \\ \left\{\!\begin{aligned} & A+B=1 \\ & 2A-2B=1 \end{aligned}\right. \ \ \ \left\{\!\begin{aligned} & A=1-B \\ & 2-2B-2B=1 \end{aligned}\right. \ \ \ \left\{\!\begin{aligned} & A = 1-B \\ & -4B=-1 \end{aligned}\right.$

$\ \left\{\!\begin{aligned} & A = 1-\frac{1}{4}=\frac{3}{4} \\ & B=\frac{1}{4} \end{aligned}$


$(*) = \int {( \frac{\frac{3}{4} }{x-2} + \frac{\frac{1}{4}}{x+2})} \, dx = \\ \\ = \frac{3}{4} \int {\frac{\, dx}{x-2}} + \frac{1}{4} \int {\frac{ \, dx }{x+2}}= \\\\=\frac{3}{4}\cdot \ln{|x-2|}+\frac{1}{4} \cdot \ln{|x+2|}+C$

Answer 2

Решение в файле................