Integral de $$$\frac{1}{x^{2} \sqrt{a^{2} - x^{2}}}$$$ em relação a $$$x$$$

Encontre $$$\int \frac{1}{x^{2} \sqrt{a^{2} - x^{2}}}\, dx$$$.

Seja $$$x=\sin{\left(u \right)} \left|{a}\right|$$$.

Então $$$dx=\left(\sin{\left(u \right)} \left|{a}\right|\right)^{\prime }du = \cos{\left(u \right)} \left|{a}\right| du$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$u=\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}$$$.

O integrando torna-se

$$$\frac{1}{x^{2} \sqrt{a^{2} - x^{2}}} = \frac{1}{a^{2} \sqrt{- a^{2} \sin^{2}{\left( u \right)} + a^{2}} \sin^{2}{\left( u \right)}}$$$

Use a identidade $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

$$$\frac{1}{a^{2} \sqrt{- a^{2} \sin^{2}{\left( u \right)} + a^{2}} \sin^{2}{\left( u \right)}}=\frac{1}{a^{2} \sqrt{1 - \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)} \left|{a}\right|}=\frac{1}{a^{2} \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)} \left|{a}\right|}$$$

Supondo que $$$\cos{\left( u \right)} \ge 0$$$, obtemos o seguinte:

$$$\frac{1}{a^{2} \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)} \left|{a}\right|} = \frac{1}{a^{2} \sin^{2}{\left( u \right)} \cos{\left( u \right)} \left|{a}\right|}$$$

A integral pode ser reescrita como

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

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

$${\color{red}{\int{\frac{1}{a^{2} \sin^{2}{\left(u \right)}} d u}}} = {\color{red}{\frac{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}{a^{2}}}}$$

Reescreva o integrando em termos da cossecante:

$$\frac{{\color{red}{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}}}{a^{2}} = \frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{a^{2}}$$

A integral de $$$\csc^{2}{\left(u \right)}$$$ é $$$\int{\csc^{2}{\left(u \right)} d u} = - \cot{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{a^{2}} = \frac{{\color{red}{\left(- \cot{\left(u \right)}\right)}}}{a^{2}}$$

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

$$- \frac{\cot{\left({\color{red}{u}} \right)}}{a^{2}} = - \frac{\cot{\left({\color{red}{\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)}}{a^{2}}$$

Portanto,

$$\int{\frac{1}{x^{2} \sqrt{a^{2} - x^{2}}} d x} = - \frac{\sqrt{1 - \frac{x^{2}}{a^{2}}} \left|{a}\right|}{a^{2} x}$$

Simplifique:

$$\int{\frac{1}{x^{2} \sqrt{a^{2} - x^{2}}} d x} = - \frac{\sqrt{a^{2} - x^{2}}}{a^{2} x}$$

Adicione a constante de integração:

$$\int{\frac{1}{x^{2} \sqrt{a^{2} - x^{2}}} d x} = - \frac{\sqrt{a^{2} - x^{2}}}{a^{2} x}+C$$

$$$\int \frac{1}{x^{2} \sqrt{a^{2} - x^{2}}}\, dx = - \frac{\sqrt{a^{2} - x^{2}}}{a^{2} x} + C$$$A