Integral de $$$\operatorname{acosh}{\left(x \right)}$$$

Halla $$$\int \operatorname{acosh}{\left(x \right)}\, dx$$$.

Para la integral $$$\int{\operatorname{acosh}{\left(x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\operatorname{acosh}{\left(x \right)}$$$ y $$$\operatorname{dv}=dx$$$.

Entonces $$$\operatorname{du}=\left(\operatorname{acosh}{\left(x \right)}\right)^{\prime }dx=\frac{1}{\sqrt{x - 1} \sqrt{x + 1}} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{1 d x}=x$$$ (los pasos pueden verse »).

Por lo tanto,

$${\color{red}{\int{\operatorname{acosh}{\left(x \right)} d x}}}={\color{red}{\left(\operatorname{acosh}{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{\sqrt{x - 1} \sqrt{x + 1}} d x}\right)}}={\color{red}{\left(x \operatorname{acosh}{\left(x \right)} - \int{\frac{x}{\sqrt{x^{2} - 1}} d x}\right)}}$$

Sea $$$u=x^{2} - 1$$$.

Entonces $$$du=\left(x^{2} - 1\right)^{\prime }dx = 2 x dx$$$ (los pasos pueden verse »), y obtenemos que $$$x dx = \frac{du}{2}$$$.

Entonces,

$$x \operatorname{acosh}{\left(x \right)} - {\color{red}{\int{\frac{x}{\sqrt{x^{2} - 1}} d x}}} = x \operatorname{acosh}{\left(x \right)} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$:

$$x \operatorname{acosh}{\left(x \right)} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}} = x \operatorname{acosh}{\left(x \right)} - {\color{red}{\left(\frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)}}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=- \frac{1}{2}$$$:

$$x \operatorname{acosh}{\left(x \right)} - \frac{{\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{2}=x \operatorname{acosh}{\left(x \right)} - \frac{{\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{2}=x \operatorname{acosh}{\left(x \right)} - \frac{{\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}=x \operatorname{acosh}{\left(x \right)} - \frac{{\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{2}=x \operatorname{acosh}{\left(x \right)} - \frac{{\color{red}{\left(2 \sqrt{u}\right)}}}{2}$$

Recordemos que $$$u=x^{2} - 1$$$:

$$x \operatorname{acosh}{\left(x \right)} - \sqrt{{\color{red}{u}}} = x \operatorname{acosh}{\left(x \right)} - \sqrt{{\color{red}{\left(x^{2} - 1\right)}}}$$

Por lo tanto,

$$\int{\operatorname{acosh}{\left(x \right)} d x} = x \operatorname{acosh}{\left(x \right)} - \sqrt{x^{2} - 1}$$

Añade la constante de integración:

$$\int{\operatorname{acosh}{\left(x \right)} d x} = x \operatorname{acosh}{\left(x \right)} - \sqrt{x^{2} - 1}+C$$

$$$\int \operatorname{acosh}{\left(x \right)}\, dx = \left(x \operatorname{acosh}{\left(x \right)} - \sqrt{x^{2} - 1}\right) + C$$$A