Lời giải:
a) ĐKXĐ: \(\left\{\begin{matrix} x+1\neq 0\\ x-1\neq 0\\ 2-2x^2\neq 0\end{matrix}\right.\Leftrightarrow x\neq \pm 1\)
b)
\(A=\left[\frac{x(x-1)}{(x-1)(x+1)}+\frac{x+1}{(x+1)(x-1)}+\frac{2x}{(x-1)(x+1)}\right].\frac{1}{x+1}=\frac{x^2+2x+1}{(x-1)(x+1)}.\frac{1}{x+1}\)
\(=\frac{(x+1)^2}{(x-1)(x+1)}.\frac{1}{x+1}=\frac{1}{x-1}\)
Để $A$ nguyên thì $1\vdots x-1$
$\Rightarrow x-1\in\left\{\pm 1\right\}$
$\Rightarrow x\in\left\{0;2\right\}$ (đều thỏa mãn đkxđ)
a) ĐKXĐ: \(x\notin\left\{1;-1\right\}\)
Ta có: \(A=\left(\dfrac{x}{x+1}+\dfrac{1}{x-1}-\dfrac{4x}{2-2x^2}\right):\left(x+1\right)\)
\(=\left(\dfrac{2x\left(x-1\right)}{2\left(x+1\right)\left(x-1\right)}+\dfrac{2\left(x+1\right)}{2\left(x+1\right)\left(x-1\right)}+\dfrac{4x}{2\left(x+1\right)\left(x-1\right)}\right)\cdot\dfrac{1}{x+1}\)
\(=\dfrac{2x^2-2x+2x+2+4x}{2\left(x+1\right)\left(x-1\right)}\cdot\dfrac{1}{x+1}\)
\(=\dfrac{2x^2+4x+2}{2\left(x+1\right)\left(x-1\right)}\cdot\dfrac{1}{x+1}\)
\(=\dfrac{2\left(x^2+2x+1\right)}{2\left(x+1\right)\left(x-1\right)}\cdot\dfrac{1}{x+1}\)
\(=\dfrac{2\left(x+1\right)^2}{2\left(x+1\right)^2\cdot\left(x-1\right)}\)
\(=\dfrac{1}{x-1}\)
b) Để A nguyên thì \(1⋮x-1\)
\(\Leftrightarrow x-1\inƯ\left(1\right)\)
\(\Leftrightarrow x-1\in\left\{1;-1\right\}\)
hay \(x\in\left\{2;0\right\}\)
Kết hợp ĐKXĐ, ta được: \(x\in\left\{2;0\right\}\)
Vậy: Để A nguyên thì \(x\in\left\{2;0\right\}\)
