Integral of $$$\sqrt{- a^{2} + u^{2}}$$$ with respect to $$$u$$$

Find $$$\int \sqrt{- a^{2} + u^{2}}\, du$$$.

Let $$$u=\cosh{\left(v \right)} \left|{a}\right|$$$.

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

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

Therefore,

$$$\sqrt{- a^{2} + u^{2}} = \sqrt{a^{2} \cosh^{2}{\left( v \right)} - a^{2}}$$$

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

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

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

$$$\sqrt{\sinh^{2}{\left( v \right)}} \left|{a}\right| = \sinh{\left( v \right)} \left|{a}\right|$$$

Thus,

$${\color{red}{\int{\sqrt{- a^{2} + u^{2}} d u}}} = {\color{red}{\int{a^{2} \sinh^{2}{\left(v \right)} d v}}}$$

Apply the power reducing formula $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$ with $$$\alpha= v $$$:

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

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

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

Expand the expression:

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

Integrate term by term:

$$\frac{{\color{red}{\int{\left(a^{2} \cosh{\left(2 v \right)} - a^{2}\right)d v}}}}{2} = \frac{{\color{red}{\left(- \int{a^{2} d v} + \int{a^{2} \cosh{\left(2 v \right)} d v}\right)}}}{2}$$

Apply the constant rule $$$\int c\, dv = c v$$$ with $$$c=a^{2}$$$:

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

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

$$- \frac{a^{2} v}{2} + \frac{{\color{red}{\int{a^{2} \cosh{\left(2 v \right)} d v}}}}{2} = - \frac{a^{2} v}{2} + \frac{{\color{red}{a^{2} \int{\cosh{\left(2 v \right)} d v}}}}{2}$$

Let $$$w=2 v$$$.

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

So,

$$- \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\int{\cosh{\left(2 v \right)} d v}}}}{2} = - \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\int{\frac{\cosh{\left(w \right)}}{2} d w}}}}{2}$$

Apply the constant multiple rule $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(w \right)} = \cosh{\left(w \right)}$$$:

$$- \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\int{\frac{\cosh{\left(w \right)}}{2} d w}}}}{2} = - \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\left(\frac{\int{\cosh{\left(w \right)} d w}}{2}\right)}}}{2}$$

The integral of the hyperbolic cosine is $$$\int{\cosh{\left(w \right)} d w} = \sinh{\left(w \right)}$$$:

$$- \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\int{\cosh{\left(w \right)} d w}}}}{4} = - \frac{a^{2} v}{2} + \frac{a^{2} {\color{red}{\sinh{\left(w \right)}}}}{4}$$

Recall that $$$w=2 v$$$:

$$- \frac{a^{2} v}{2} + \frac{a^{2} \sinh{\left({\color{red}{w}} \right)}}{4} = - \frac{a^{2} v}{2} + \frac{a^{2} \sinh{\left({\color{red}{\left(2 v\right)}} \right)}}{4}$$

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

$$\frac{a^{2} \sinh{\left(2 {\color{red}{v}} \right)}}{4} - \frac{a^{2} {\color{red}{v}}}{2} = \frac{a^{2} \sinh{\left(2 {\color{red}{\operatorname{acosh}{\left(\frac{u}{\left|{a}\right|} \right)}}} \right)}}{4} - \frac{a^{2} {\color{red}{\operatorname{acosh}{\left(\frac{u}{\left|{a}\right|} \right)}}}}{2}$$

Therefore,

$$\int{\sqrt{- a^{2} + u^{2}} d u} = \frac{a^{2} \sinh{\left(2 \operatorname{acosh}{\left(\frac{u}{\left|{a}\right|} \right)} \right)}}{4} - \frac{a^{2} \operatorname{acosh}{\left(\frac{u}{\left|{a}\right|} \right)}}{2}$$

Using the formulas $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, simplify the expression:

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

Add the constant of integration:

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

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