Integral de $$$\operatorname{atan}{\left(2 x \right)}$$$

Halla $$$\int \operatorname{atan}{\left(2 x \right)}\, dx$$$.

Sea $$$u=2 x$$$.

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

La integral se convierte en

$${\color{red}{\int{\operatorname{atan}{\left(2 x \right)} d x}}} = {\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{2} d u}}}$$

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 $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\operatorname{atan}{\left(u \right)} d u}}{2}\right)}}$$

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

Sean $$$\operatorname{\mu}=\operatorname{atan}{\left(u \right)}$$$ y $$$\operatorname{dv}=du$$$.

Entonces $$$\operatorname{d\mu}=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du=\frac{du}{u^{2} + 1}$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{1 d u}=u$$$ (los pasos pueden verse »).

Entonces,

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

Sea $$$v=u^{2} + 1$$$.

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

La integral se convierte en

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

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

$$\frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{2 v} d v}}}}{2} = \frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{{\color{red}{\left(\frac{\int{\frac{1}{v} d v}}{2}\right)}}}{2}$$

La integral de $$$\frac{1}{v}$$$ es $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$\frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{4} = \frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{4}$$

Recordemos que $$$v=u^{2} + 1$$$:

$$\frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{4} = \frac{u \operatorname{atan}{\left(u \right)}}{2} - \frac{\ln{\left(\left|{{\color{red}{\left(u^{2} + 1\right)}}}\right| \right)}}{4}$$

Recordemos que $$$u=2 x$$$:

$$- \frac{\ln{\left(1 + {\color{red}{u}}^{2} \right)}}{4} + \frac{{\color{red}{u}} \operatorname{atan}{\left({\color{red}{u}} \right)}}{2} = - \frac{\ln{\left(1 + {\color{red}{\left(2 x\right)}}^{2} \right)}}{4} + \frac{{\color{red}{\left(2 x\right)}} \operatorname{atan}{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

Por lo tanto,

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

Añade la constante de integración:

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

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