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

Find $$$\int \left(- \frac{1}{u \sqrt{1 - u^{2}}}\right)\, du$$$.

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

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

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

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

The integral becomes

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

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

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

Let $$$v=\cosh{\left(w \right)}$$$.

Then $$$dv=\left(\cosh{\left(w \right)}\right)^{\prime }dw = \sinh{\left(w \right)} dw$$$ (steps can be seen »).

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

Therefore,

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

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

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

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

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

Thus,

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

Apply the constant rule $$$\int c\, dw = c w$$$ with $$$c=1$$$:

$${\color{red}{\int{1 d w}}} = {\color{red}{w}}$$

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

$${\color{red}{w}} = {\color{red}{\operatorname{acosh}{\left(v \right)}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int \left(- \frac{1}{u \sqrt{1 - u^{2}}}\right)\, du = \operatorname{acosh}{\left(\frac{1}{u} \right)} + C$$$A