Integral of $$$\frac{z \operatorname{asin}{\left(2 x \right)}}{20}$$$ with respect to $$$x$$$

Find $$$\int \frac{z \operatorname{asin}{\left(2 x \right)}}{20}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{z}{20}$$$ and $$$f{\left(x \right)} = \operatorname{asin}{\left(2 x \right)}$$$:

$${\color{red}{\int{\frac{z \operatorname{asin}{\left(2 x \right)}}{20} d x}}} = {\color{red}{\left(\frac{z \int{\operatorname{asin}{\left(2 x \right)} d x}}{20}\right)}}$$

Let $$$u=2 x$$$.

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

Thus,

$$\frac{z {\color{red}{\int{\operatorname{asin}{\left(2 x \right)} d x}}}}{20} = \frac{z {\color{red}{\int{\frac{\operatorname{asin}{\left(u \right)}}{2} d u}}}}{20}$$

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

$$\frac{z {\color{red}{\int{\frac{\operatorname{asin}{\left(u \right)}}{2} d u}}}}{20} = \frac{z {\color{red}{\left(\frac{\int{\operatorname{asin}{\left(u \right)} d u}}{2}\right)}}}{20}$$

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

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

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

So,

$$\frac{z {\color{red}{\int{\operatorname{asin}{\left(u \right)} d u}}}}{40}=\frac{z {\color{red}{\left(\operatorname{asin}{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{\sqrt{1 - u^{2}}} d u}\right)}}}{40}=\frac{z {\color{red}{\left(u \operatorname{asin}{\left(u \right)} - \int{\frac{u}{\sqrt{1 - u^{2}}} d u}\right)}}}{40}$$

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

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

The integral becomes

$$\frac{z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\int{\frac{u}{\sqrt{1 - u^{2}}} d u}}}\right)}{40} = \frac{z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\int{\left(- \frac{1}{2 \sqrt{v}}\right)d v}}}\right)}{40}$$

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

$$\frac{z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\int{\left(- \frac{1}{2 \sqrt{v}}\right)d v}}}\right)}{40} = \frac{z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\left(- \frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}\right)}{40}$$

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

$$\frac{z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{2}\right)}{40}=\frac{z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{2}\right)}{40}=\frac{z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}\right)}{40}=\frac{z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{2}\right)}{40}=\frac{z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\left(2 \sqrt{v}\right)}}}{2}\right)}{40}$$

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

$$\frac{z \left(u \operatorname{asin}{\left(u \right)} + \sqrt{{\color{red}{v}}}\right)}{40} = \frac{z \left(u \operatorname{asin}{\left(u \right)} + \sqrt{{\color{red}{\left(1 - u^{2}\right)}}}\right)}{40}$$

Recall that $$$u=2 x$$$:

$$\frac{z \left(\sqrt{1 - {\color{red}{u}}^{2}} + {\color{red}{u}} \operatorname{asin}{\left({\color{red}{u}} \right)}\right)}{40} = \frac{z \left(\sqrt{1 - {\color{red}{\left(2 x\right)}}^{2}} + {\color{red}{\left(2 x\right)}} \operatorname{asin}{\left({\color{red}{\left(2 x\right)}} \right)}\right)}{40}$$

Therefore,

$$\int{\frac{z \operatorname{asin}{\left(2 x \right)}}{20} d x} = \frac{z \left(2 x \operatorname{asin}{\left(2 x \right)} + \sqrt{1 - 4 x^{2}}\right)}{40}$$

Add the constant of integration:

$$\int{\frac{z \operatorname{asin}{\left(2 x \right)}}{20} d x} = \frac{z \left(2 x \operatorname{asin}{\left(2 x \right)} + \sqrt{1 - 4 x^{2}}\right)}{40}+C$$

$$$\int \frac{z \operatorname{asin}{\left(2 x \right)}}{20}\, dx = \frac{z \left(2 x \operatorname{asin}{\left(2 x \right)} + \sqrt{1 - 4 x^{2}}\right)}{40} + C$$$A