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

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

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

Then $$$dx=\left(8 \sin{\left(u \right)}\right)^{\prime }du = 8 \cos{\left(u \right)} du$$$ (steps can be seen »).

Also, it follows that $$$u=\operatorname{asin}{\left(\frac{x}{8} \right)}$$$.

Thus,

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

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

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

Assuming that $$$\cos{\left( u \right)} \ge 0$$$, we obtain the following:

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

Therefore,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{64}$$$ and $$$f{\left(u \right)} = \frac{1}{\sin^{2}{\left(u \right)}}$$$:

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

Rewrite the integrand in terms of the cosecant:

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

The integral of $$$\csc^{2}{\left(u \right)}$$$ is $$$\int{\csc^{2}{\left(u \right)} d u} = - \cot{\left(u \right)}$$$:

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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