For the integral ∫asin(x)dx, use integration by parts ∫udv=uv−∫vdu.
Let u=asin(x) and dv=dx.
Then du=(asin(x))′dx=[tex]\rm \dfrac{dx}{ \sqrt{1 - {x}^{2} } } [/tex] and v=∫1dx=x
So,
[tex] \rm{\int{\operatorname{asin}{\left(x \right)} d x}}=\color{h}{\left(\operatorname{asin}{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{\sqrt{1 - x^{2}}} d x}\right)}=\color{h}{\left(x \operatorname{asin}{\left(x \right)} - \int{\frac{x}{\sqrt{1 - x^{2}}} d x}\right)} \\ [/tex]
Let u=1−x2.
Then du=(1−x2)′dx=−2xdx and we have that xdx=−du/2.
The integral can be rewritten as
[tex] \rm x \operatorname{asin}{\left(x \right)} - \color{g}{\int{\frac{x}{\sqrt{1 - x^{2}}} d x}} = x \operatorname{asin}{\left(x \right)} - \color{j}{\int{\left(- \frac{1}{2 \sqrt{u}}\right)d u}} \\ [/tex]
Apply the constant multiple rule ∫cf(u)du=c∫f(u)du with c=-1/2 and f(u)=[tex] \frac{1}{ \sqrt{u} } [/tex]
[tex] \rm x \operatorname{asin}{\left(x \right)} - \color{h}{\int{\left(- \frac{1}{2 \sqrt{u}}\right)d u}} = x \operatorname{asin}{\left(x \right)} - \color{h}{\left(- \frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)} \\ [/tex]
Apply the power rule
[tex] \rm\int u^{n}\, du = \frac{u^{n + 1}}{n + 1} \\ [/tex]
with n=−1/2
[tex] \rm x \operatorname{asin}{\left(x \right)} + \frac{\color{h}{\int{\frac{1}{\sqrt{u}} d u}}}{2}=x \operatorname{asin}{\left(x \right)} + \frac{\color{j}{\int{u^{- \frac{1}{2}} d u}}}{2}=x \operatorname{asin}{\left(x \right)} + \frac{\color{j}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}{2}=x \operatorname{asin}{\left(x \right)} + \frac{\color{j}{\left(2 u^{\frac{1}{2}}\right)}}{2}=x \operatorname{asin}{\left(x \right)} + \frac{\color{h}{\left(2 \sqrt{u}\right)}}{2} \\ [/tex]
[tex] \rm Recall \: that \: u=1− {x}^{2} [/tex]
[tex] \rm x \operatorname{asin}{\left(x \right)} + \sqrt{\color{re}{u}} = x \operatorname{asin}{\left(x \right)} + \sqrt{\color{rd}{\left(1 - x^{2}\right)}} \\ [/tex]
Therefore,
[tex] \rm\int{\operatorname{asin}{\left(x \right)} d x} = x \operatorname{asin}{\left(x \right)} + \sqrt{1 - x^{2}}+C \\ [/tex]
