Integral de $$$\frac{1}{x^{2} \sqrt{64 - x^{2}}}$$$

Halla $$$\int \frac{1}{x^{2} \sqrt{64 - x^{2}}}\, dx$$$.

Sea $$$x=8 \sin{\left(u \right)}$$$.

Entonces $$$dx=\left(8 \sin{\left(u \right)}\right)^{\prime }du = 8 \cos{\left(u \right)} du$$$ (los pasos pueden verse »).

Además, se sigue que $$$u=\operatorname{asin}{\left(\frac{x}{8} \right)}$$$.

Por lo tanto,

$$$\frac{1}{x^{2} \sqrt{64 - x^{2}}} = \frac{1}{64 \sqrt{64 - 64 \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}$$$

Utiliza la identidad $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

$$$\frac{1}{64 \sqrt{64 - 64 \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}=\frac{1}{512 \sqrt{1 - \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}=\frac{1}{512 \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}$$$

Suponiendo que $$$\cos{\left( u \right)} \ge 0$$$, obtenemos lo siguiente:

$$$\frac{1}{512 \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}} = \frac{1}{512 \sin^{2}{\left( u \right)} \cos{\left( u \right)}}$$$

Por lo tanto,

$${\color{red}{\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x}}} = {\color{red}{\int{\frac{1}{64 \sin^{2}{\left(u \right)}} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{64}$$$ y $$$f{\left(u \right)} = \frac{1}{\sin^{2}{\left(u \right)}}$$$:

$${\color{red}{\int{\frac{1}{64 \sin^{2}{\left(u \right)}} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}{64}\right)}}$$

Reescribe el integrando en términos de la cosecante:

$$\frac{{\color{red}{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}}}{64} = \frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{64}$$

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

$$\frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{64} = \frac{{\color{red}{\left(- \cot{\left(u \right)}\right)}}}{64}$$

Recordemos que $$$u=\operatorname{asin}{\left(\frac{x}{8} \right)}$$$:

$$- \frac{\cot{\left({\color{red}{u}} \right)}}{64} = - \frac{\cot{\left({\color{red}{\operatorname{asin}{\left(\frac{x}{8} \right)}}} \right)}}{64}$$

Por lo tanto,

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{1 - \frac{x^{2}}{64}}}{8 x}$$

Simplificar:

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{64 - x^{2}}}{64 x}$$

Añade la constante de integración:

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{64 - x^{2}}}{64 x}+C$$

$$$\int \frac{1}{x^{2} \sqrt{64 - x^{2}}}\, dx = - \frac{\sqrt{64 - x^{2}}}{64 x} + C$$$A