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

Encontre $$$\int \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{x}}\, dx$$$.

Seja $$$u=\sqrt{x}$$$.

Então $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (veja os passos »), e obtemos $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Portanto,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:

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

Para a integral $$$\int{\operatorname{atan}{\left(u \right)} d u}$$$, use integração por partes $$$\int \operatorname{\kappa} \operatorname{dv} = \operatorname{\kappa}\operatorname{v} - \int \operatorname{v} \operatorname{d\kappa}$$$.

Sejam $$$\operatorname{\kappa}=\operatorname{atan}{\left(u \right)}$$$ e $$$\operatorname{dv}=du$$$.

Então $$$\operatorname{d\kappa}=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du=\frac{du}{u^{2} + 1}$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{1 d u}=u$$$ (os passos podem ser vistos »).

A integral torna-se

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

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

Então $$$dv=\left(u^{2} + 1\right)^{\prime }du = 2 u du$$$ (veja os passos »), e obtemos $$$u du = \frac{dv}{2}$$$.

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(v \right)} = \frac{1}{v}$$$:

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

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

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

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

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

Recorde que $$$u=\sqrt{x}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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