In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle …

Use the common integral: ∫ cosh (u) d u = sinh (u) = 2 1 sinh (u) Substitute back u = 2 t = 2 1 sinh ( 2 t ) Add a constant to the solution = 2 1 sinh ( 2 t ) + C

基本知识： \cosh x=\frac {e^x+e^ {-x}}2,\sinh x=\frac {e^x-e^ {-x}}2,\tanh x=\frac {e^x-e^ {-x}} {e^x+e^ {-x}} \displaystyle {\rm arsinh } \ x=\ln (x+\sqrt {x^2+1}), {\rm arcosh } \ x=\ln (x+\sqrt …

2024年12月30日 · 主要是 \sqrt {x^2-a^2},\sqrt {x^2+a^2}。 比如说， \sqrt {x^2+1}dx,可以利用双曲函数的基本恒等式，令 x=sinh (t) \sqrt {x^2+1}dt=cosh (t)d (sinht)= (cosh (t))^2 dt=\frac …

$\text{cosh}\ 2x = \text{cosh}^2x + \text{sinh}^2x= 2 \text{cosh}^2x - 1 = 1 + 2 \text{sinh}^2x$ $\text{tanh}\ 2x = \frac{2\text{tanh}\ x}{1 + \text{tanh}^2x}$ Half angle formulas

\color {blue} {\textbf {双曲函数的定义}} 1、 双曲正弦. \sinh x = \frac {e^ {x} - e^ {-x}} {2} \\ 2、 双曲余弦. \cosh x = \frac {e^ {x} + e^ {-x}} {2} \\ 3、 双曲正切. \sinh x =\frac {\sinh} {\cosh} = \frac …

双曲三角函数把实数轴映射到了半个双曲线上，为什么我们不看看这组函数会把虚轴映射到哪里呢？ 设 t=i\alpha,\alpha\in\mathbb R 则有： \cosh^2 (i\alpha)-\sinh^2 (i\alpha)=1. 由于 \overline …

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