Integral of $$$\frac{1}{x \sqrt{25 - x^{2}}}$$$

Find $$$\int \frac{1}{x \sqrt{25 - x^{2}}}\, dx$$$.

Let $$$u=\frac{1}{x}$$$.

Then $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (steps can be seen »), and we have that $$$\frac{dx}{x^{2}} = - du$$$.

Thus,

$${\color{red}{\int{\frac{1}{x \sqrt{25 - x^{2}}} d x}}} = {\color{red}{\int{\left(- \frac{1}{\sqrt{25 u^{2} - 1}}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{1}{\sqrt{25 u^{2} - 1}}$$$:

$${\color{red}{\int{\left(- \frac{1}{\sqrt{25 u^{2} - 1}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{25 u^{2} - 1}} d u}\right)}}$$

Let $$$u=\frac{\cosh{\left(v \right)}}{5}$$$.

Then $$$du=\left(\frac{\cosh{\left(v \right)}}{5}\right)^{\prime }dv = \frac{\sinh{\left(v \right)}}{5} dv$$$ (steps can be seen »).

Also, it follows that $$$v=\operatorname{acosh}{\left(5 u \right)}$$$.

Thus,

$$$\frac{1}{\sqrt{25 u ^{2} - 1}} = \frac{1}{\sqrt{\cosh^{2}{\left( v \right)} - 1}}$$$

Use the identity $$$\cosh^{2}{\left( v \right)} - 1 = \sinh^{2}{\left( v \right)}$$$:

$$$\frac{1}{\sqrt{\cosh^{2}{\left( v \right)} - 1}}=\frac{1}{\sqrt{\sinh^{2}{\left( v \right)}}}$$$

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

$$$\frac{1}{\sqrt{\sinh^{2}{\left( v \right)}}} = \frac{1}{\sinh{\left( v \right)}}$$$

Therefore,

$$- {\color{red}{\int{\frac{1}{\sqrt{25 u^{2} - 1}} d u}}} = - {\color{red}{\int{\frac{1}{5} d v}}}$$

Apply the constant rule $$$\int c\, dv = c v$$$ with $$$c=\frac{1}{5}$$$:

$$- {\color{red}{\int{\frac{1}{5} d v}}} = - {\color{red}{\left(\frac{v}{5}\right)}}$$

Recall that $$$v=\operatorname{acosh}{\left(5 u \right)}$$$:

$$- \frac{{\color{red}{v}}}{5} = - \frac{{\color{red}{\operatorname{acosh}{\left(5 u \right)}}}}{5}$$

Recall that $$$u=\frac{1}{x}$$$:

$$- \frac{\operatorname{acosh}{\left(5 {\color{red}{u}} \right)}}{5} = - \frac{\operatorname{acosh}{\left(5 {\color{red}{\frac{1}{x}}} \right)}}{5}$$

Therefore,

$$\int{\frac{1}{x \sqrt{25 - x^{2}}} d x} = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5}$$

Add the constant of integration:

$$\int{\frac{1}{x \sqrt{25 - x^{2}}} d x} = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5}+C$$

$$$\int \frac{1}{x \sqrt{25 - x^{2}}}\, dx = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5} + C$$$A