Integral of arcsin(x)*arcsin(x) dx

1. Use integration by parts:

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

   Let $u{\left(x \right)} = \operatorname{asin}^{2}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$.

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

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

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

      $$\int 1\, dx = x$$

   Now evaluate the sub-integral.

2. Use integration by parts:

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

   Let $u{\left(x \right)} = 2 \operatorname{asin}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{x}{\sqrt{1 - x^{2}}}$.

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

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

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

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

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

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

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

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

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

         So, the result is: $- \sqrt{u}$

      Now substitute $u$ back in:

      $$- \sqrt{1 - x^{2}}$$

   Now evaluate the sub-integral.

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

   $$\int \left(-2\right)\, dx = - 2 x$$

4. Add the constant of integration:

   $$x \operatorname{asin}^{2}{\left(x \right)} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}+ \mathrm{constant}$$

The answer is: