Integral de $$$\frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}}$$$

Encontre $$$\int \frac{1}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}}\, dx$$$.

Seja $$$u=\operatorname{asin}{\left(x \right)}$$$.

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

Assim,

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

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

$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recorde que $$$u=\operatorname{asin}{\left(x \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\operatorname{asin}{\left(x \right)}}}}\right| \right)}$$

Portanto,

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

Adicione a constante de integração:

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

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