Integral of (1/2sqrt(x))arcsinsqrt(x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sqrt{x}$.

      Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $du$:

      $$\int u^{2} \operatorname{asin}{\left(u \right)}\, du$$

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(u \right)} = \operatorname{asin}{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = u^{2}$.

         Then $\operatorname{du}{\left(u \right)} = \frac{1}{\sqrt{1 - u^{2}}}$.

         To find $v{\left(u \right)}$:

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int u^{2}\, du = \frac{u^{3}}{3}$$

         Now evaluate the sub-integral.

      2. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{u^{3}}{3 \sqrt{1 - u^{2}}}\, du = \frac{\int \frac{u^{3}}{\sqrt{1 - u^{2}}}\, du}{3}$$

         TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**3, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), ConstantRule(constant=-1, context=-1, symbol=_u)], context=_u**2 - 1, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), context=sin(_theta)**3, symbol=_theta), restriction=(_u > -1) & (_u < 1), context=_u**3/sqrt(1 - _u**2), symbol=_u)

         So, the result is: $\frac{\begin{cases} \frac{\left(1 - u^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{1 - u^{2}} & \text{for}\: u > -1 \wedge u < 1 \end{cases}}{3}$

      Now substitute $u$ back in:

      $$\frac{x^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}}{3} - \frac{\begin{cases} \frac{\left(1 - x\right)^{\frac{3}{2}}}{3} - \sqrt{1 - x} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{3}$$

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = \operatorname{asin}{\left(\sqrt{x} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{\sqrt{x}}{2}$.

      Then $\operatorname{du}{\left(x \right)} = \frac{1}{2 \sqrt{x} \sqrt{1 - x}}$.

      To find $v{\left(x \right)}$:

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \frac{\sqrt{x}}{2}\, dx = \frac{\int \sqrt{x}\, dx}{2}$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$$

         So, the result is: $\frac{x^{\frac{3}{2}}}{3}$

      Now evaluate the sub-integral.

   2. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{x}{6 \sqrt{1 - x}}\, dx = \frac{\int \frac{x}{\sqrt{1 - x}}\, dx}{6}$$

      1. Let $u = \frac{1}{\sqrt{1 - x}}$.

         Then let $du = \frac{dx}{2 \left(1 - x\right)^{\frac{3}{2}}}$ and substitute $du$:

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

         1. Integrate term-by-term:

            1. The integral of a constant times a function is the constant times the integral of the function:

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

               1. There are multiple ways to do this integral.

                  Method #1

                  1. Rewrite the integrand:

                     $$\left(1 - \frac{1}{u^{2}}\right)^{2} = 1 - \frac{2}{u^{2}} + \frac{1}{u^{4}}$$

                  2. Integrate term-by-term:

                     1. The integral of a constant is the constant times the variable of integration:

                        $$\int 1\, du = u$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

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

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$$

                        So, the result is: $\frac{2}{u}$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$$

                     The result is: $u + \frac{2}{u} - \frac{1}{3 u^{3}}$

                  Method #2

                  1. Rewrite the integrand:

                     $$\left(1 - \frac{1}{u^{2}}\right)^{2} = \frac{u^{4} - 2 u^{2} + 1}{u^{4}}$$

                  2. Rewrite the integrand:

                     $$\frac{u^{4} - 2 u^{2} + 1}{u^{4}} = 1 - \frac{2}{u^{2}} + \frac{1}{u^{4}}$$

                  3. Integrate term-by-term:

                     1. The integral of a constant is the constant times the variable of integration:

                        $$\int 1\, du = u$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

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

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$$

                        So, the result is: $\frac{2}{u}$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$$

                     The result is: $u + \frac{2}{u} - \frac{1}{3 u^{3}}$

               So, the result is: $- 2 u - \frac{4}{u} + \frac{2}{3 u^{3}}$

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 2\, du = 2 u$$

            1. The integral of a constant times a function is the constant times the integral of the function:

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

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$$

               So, the result is: $\frac{2}{u}$

            The result is: $- \frac{2}{u} + \frac{2}{3 u^{3}}$

         Now substitute $u$ back in:

         $$\frac{2 \left(1 - x\right)^{\frac{3}{2}}}{3} - 2 \sqrt{1 - x}$$

      So, the result is: $\frac{\left(1 - x\right)^{\frac{3}{2}}}{9} - \frac{\sqrt{1 - x}}{3}$

2. Now simplify:

   $$\begin{cases} \frac{x^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}}{3} + \frac{\sqrt{1 - x} \left(x + 2\right)}{9} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}$$

3. Add the constant of integration:

   $$\begin{cases} \frac{x^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}}{3} + \frac{\sqrt{1 - x} \left(x + 2\right)}{9} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}+ \mathrm{constant}$$

The answer is:

$$\begin{cases} \frac{x^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}}{3} + \frac{\sqrt{1 - x} \left(x + 2\right)}{9} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}+ \mathrm{constant}$$