Integral de $$$\frac{1}{x \sqrt{x^{2} + 4 x + 1}}$$$

Halla $$$\int \frac{1}{x \sqrt{x^{2} + 4 x + 1}}\, dx$$$.

Sea $$$u=\frac{1}{x}$$$.

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

La integral puede reescribirse como

$${\color{red}{\int{\frac{1}{x \sqrt{x^{2} + 4 x + 1}} d x}}} = {\color{red}{\int{\left(- \frac{1}{\sqrt{u^{2} + 4 u + 1}}\right)d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = \frac{1}{\sqrt{u^{2} + 4 u + 1}}$$$:

$${\color{red}{\int{\left(- \frac{1}{\sqrt{u^{2} + 4 u + 1}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{u^{2} + 4 u + 1}} d u}\right)}}$$

Completa el cuadrado (se pueden ver los pasos »): $$$ u ^{2} + 4 u + 1 = \left( u + 2\right)^{2} - 3$$$:

$$- {\color{red}{\int{\frac{1}{\sqrt{u^{2} + 4 u + 1}} d u}}} = - {\color{red}{\int{\frac{1}{\sqrt{\left(u + 2\right)^{2} - 3}} d u}}}$$

Sea $$$v=u + 2$$$.

Entonces $$$dv=\left(u + 2\right)^{\prime }du = 1 du$$$ (los pasos pueden verse »), y obtenemos que $$$du = dv$$$.

Por lo tanto,

$$- {\color{red}{\int{\frac{1}{\sqrt{\left(u + 2\right)^{2} - 3}} d u}}} = - {\color{red}{\int{\frac{1}{\sqrt{v^{2} - 3}} d v}}}$$

Sea $$$v=\sqrt{3} \cosh{\left(w \right)}$$$.

Entonces $$$dv=\left(\sqrt{3} \cosh{\left(w \right)}\right)^{\prime }dw = \sqrt{3} \sinh{\left(w \right)} dw$$$ (los pasos pueden verse »).

Además, se sigue que $$$w=\operatorname{acosh}{\left(\frac{\sqrt{3} v}{3} \right)}$$$.

El integrando se convierte en

$$$\frac{1}{\sqrt{ v ^{2} - 3}} = \frac{1}{\sqrt{3 \cosh^{2}{\left( w \right)} - 3}}$$$

Utiliza la identidad $$$\cosh^{2}{\left( w \right)} - 1 = \sinh^{2}{\left( w \right)}$$$:

$$$\frac{1}{\sqrt{3 \cosh^{2}{\left( w \right)} - 3}}=\frac{\sqrt{3}}{3 \sqrt{\cosh^{2}{\left( w \right)} - 1}}=\frac{\sqrt{3}}{3 \sqrt{\sinh^{2}{\left( w \right)}}}$$$

Suponiendo que $$$\sinh{\left( w \right)} \ge 0$$$, obtenemos lo siguiente:

$$$\frac{\sqrt{3}}{3 \sqrt{\sinh^{2}{\left( w \right)}}} = \frac{\sqrt{3}}{3 \sinh{\left( w \right)}}$$$

Por lo tanto,

$$- {\color{red}{\int{\frac{1}{\sqrt{v^{2} - 3}} d v}}} = - {\color{red}{\int{1 d w}}}$$

Aplica la regla de la constante $$$\int c\, dw = c w$$$ con $$$c=1$$$:

$$- {\color{red}{\int{1 d w}}} = - {\color{red}{w}}$$

Recordemos que $$$w=\operatorname{acosh}{\left(\frac{\sqrt{3} v}{3} \right)}$$$:

$$- {\color{red}{w}} = - {\color{red}{\operatorname{acosh}{\left(\frac{\sqrt{3} v}{3} \right)}}}$$

Recordemos que $$$v=u + 2$$$:

$$- \operatorname{acosh}{\left(\frac{\sqrt{3} {\color{red}{v}}}{3} \right)} = - \operatorname{acosh}{\left(\frac{\sqrt{3} {\color{red}{\left(u + 2\right)}}}{3} \right)}$$

Recordemos que $$$u=\frac{1}{x}$$$:

$$- \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 + {\color{red}{u}}\right)}{3} \right)} = - \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 + {\color{red}{\frac{1}{x}}}\right)}{3} \right)}$$

Por lo tanto,

$$\int{\frac{1}{x \sqrt{x^{2} + 4 x + 1}} d x} = - \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 + \frac{1}{x}\right)}{3} \right)}$$

Simplificar:

$$\int{\frac{1}{x \sqrt{x^{2} + 4 x + 1}} d x} = - \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3 x} \right)}$$

Añade la constante de integración:

$$\int{\frac{1}{x \sqrt{x^{2} + 4 x + 1}} d x} = - \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3 x} \right)}+C$$

$$$\int \frac{1}{x \sqrt{x^{2} + 4 x + 1}}\, dx = - \operatorname{acosh}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3 x} \right)} + C$$$A