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恒等变换

恒等变换

恒等变换

这里专门记录一些比较特别的恒等变换技巧

常用的恒等变换式子

逆函数

\[h(x) = g(f(x))\\ x = f^{-1} \circ g^{-1} \circ g \circ f(x)\\ h^{-1}(x) = f^{-1}(g^{-1}(x))\\\]

代数

特别注意 \(\sqrt{x^2} = (x^2)^{1/2} = |x|\)

反三角函数相关

\[\dfrac{1}{\sqrt{1+x^2} - x} = \sqrt{1+x^2} + x\] \[\dfrac{1}{x - \sqrt{x^2 - 1}} = \sqrt{x^2 - 1} + x\] \[(e^{x} + e^{-x})^2 = e^{2x} + e^{-2x} + 2\\ (e^{x} - e^{-x})^2 = e^{2x} - e^{-2x} + 2\]

特殊

\[\dfrac{1+e^x}{1+e^{-x}} = e^x\] \[(y^2-y^{-2})^2 = y^4 + y^{-4} -2 = (y^2+y^{-2})^2 - 4\]

拆分相关

\[\dfrac{x^2}{1-x^2} = \dfrac{1}{2(1-x)} + \dfrac{1}{2(1+x)} -1\]

根式转换

\[\sqrt{x + \sqrt{y}}\]

能化简的必要条件是$ x^2 - y = N^2 $, 通过待定系数法可以化简

\[\sqrt{x \pm \sqrt{y}} = \sqrt{a} \pm \sqrt{b}\] \[\begin{cases} a + b = x\\ 4ab = y \end{cases}\] \[N = \sqrt{x^2 - y}\] \[a,b = \dfrac{x \pm N}{2}\] \[\sqrt{\sqrt{z} \pm \sqrt{y}} = \sqrt{a} \pm \sqrt{b}\] \[\begin{cases} a + b = \sqrt{z}\\ 4ab = y \end{cases}\]

\(N = \sqrt{z - y}\) 8
\(a,b = \dfrac{\sqrt{z} \pm N}{2}\)

代数恒等式

\((a + b)^2 = a^2 + 2ab + b^2\) \((a - b)^2 = a^2 - 2ab + b^2\) \((ab)^2 = a^2 b^2\) \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) \((a + b)(a - b) = a^2 - b^2\) \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\) \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) \((a + b)^2 - (a - b)^2 = 4ab\) \(a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2\) \(a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)\) \((a^2 + b^2)^2 = a^4 + 2a^2b^2 + b^4\) \((a + b + c)(a + b - c) = a^2 + 2ab + b^2 - c^2\) \(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)\)

不等式

基本不等式 $几何 \leq 算数 \leq 平方$ $a>0 \quad b>0$ \(\dfrac{1}{1/a + 1/b} \leq \sqrt{ab} \leq \dfrac{a+b}{2} \leq \sqrt{\dfrac{a^2+b^2}{2}}\)

绝对值不等式

\[\begin{align*} |a\pm b| \leq |a| + |b|\\ |\int_b^af(x)dx| \leq \int_b^a|f(x)|dx\\ ||a|-|b||\leq |a-b| \end{align*}\]

分数不等式

$0<a<x<b\quad 0<c<y<d$ \(\dfrac{c}{b} < \frac{y}{x} < \frac{d}{a}\)

函数相关

\[\begin{align*} e^x \geq x+1\\ \dfrac{1}{1+x} < \ln (1+\frac{1}{x}) < \frac{1}{x} \quad x > 0 \end{align*}\]
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