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

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

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

Sejam $$$\operatorname{u}=\operatorname{atan}{\left(\sqrt{x} \right)}$$$ e $$$\operatorname{dv}=dx$$$.

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

A integral pode ser reescrita como

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

Simplifique o integrando:

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

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

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

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

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

Assim,

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

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

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

Reescreva e separe a fração:

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=1$$$:

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

A integral de $$$\frac{1}{u^{2} + 1}$$$ é $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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