Integral of $$$\operatorname{asinh}{\left(\frac{y}{2} \right)}$$$

Find $$$\int \operatorname{asinh}{\left(\frac{y}{2} \right)}\, dy$$$.

Let $$$u=\frac{y}{2}$$$.

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

Therefore,

$${\color{red}{\int{\operatorname{asinh}{\left(\frac{y}{2} \right)} d y}}} = {\color{red}{\int{2 \operatorname{asinh}{\left(u \right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=2$$$ and $$$f{\left(u \right)} = \operatorname{asinh}{\left(u \right)}$$$:

$${\color{red}{\int{2 \operatorname{asinh}{\left(u \right)} d u}}} = {\color{red}{\left(2 \int{\operatorname{asinh}{\left(u \right)} d u}\right)}}$$

For the integral $$$\int{\operatorname{asinh}{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Let $$$\operatorname{\omega}=\operatorname{asinh}{\left(u \right)}$$$ and $$$\operatorname{dv}=du$$$.

Then $$$\operatorname{d\omega}=\left(\operatorname{asinh}{\left(u \right)}\right)^{\prime }du=\frac{du}{\sqrt{u^{2} + 1}}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d u}=u$$$ (steps can be seen »).

Thus,

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

Let $$$v=u^{2} + 1$$$.

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

So,

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

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 $$$f{\left(v \right)} = \frac{1}{\sqrt{v}}$$$:

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

Apply the power rule $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{1}{2}$$$:

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

Recall that $$$v=u^{2} + 1$$$:

$$2 u \operatorname{asinh}{\left(u \right)} - 2 \sqrt{{\color{red}{v}}} = 2 u \operatorname{asinh}{\left(u \right)} - 2 \sqrt{{\color{red}{\left(u^{2} + 1\right)}}}$$

Recall that $$$u=\frac{y}{2}$$$:

$$- 2 \sqrt{1 + {\color{red}{u}}^{2}} + 2 {\color{red}{u}} \operatorname{asinh}{\left({\color{red}{u}} \right)} = - 2 \sqrt{1 + {\color{red}{\left(\frac{y}{2}\right)}}^{2}} + 2 {\color{red}{\left(\frac{y}{2}\right)}} \operatorname{asinh}{\left({\color{red}{\left(\frac{y}{2}\right)}} \right)}$$

Therefore,

$$\int{\operatorname{asinh}{\left(\frac{y}{2} \right)} d y} = y \operatorname{asinh}{\left(\frac{y}{2} \right)} - 2 \sqrt{\frac{y^{2}}{4} + 1}$$

Simplify:

$$\int{\operatorname{asinh}{\left(\frac{y}{2} \right)} d y} = y \operatorname{asinh}{\left(\frac{y}{2} \right)} - \sqrt{y^{2} + 4}$$

Add the constant of integration:

$$\int{\operatorname{asinh}{\left(\frac{y}{2} \right)} d y} = y \operatorname{asinh}{\left(\frac{y}{2} \right)} - \sqrt{y^{2} + 4}+C$$

$$$\int \operatorname{asinh}{\left(\frac{y}{2} \right)}\, dy = \left(y \operatorname{asinh}{\left(\frac{y}{2} \right)} - \sqrt{y^{2} + 4}\right) + C$$$A