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

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

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

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

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

Portanto,

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

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

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

A integral pode ser reescrita como

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

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

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

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=- \frac{1}{2}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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