Integral of x*arccos(x^2) dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \frac{\operatorname{acos}{\left(u \right)}}{2}\, du$$

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

         $$\int \operatorname{acos}{\left(u \right)}\, du = \frac{\int \operatorname{acos}{\left(u \right)}\, du}{2}$$

         1. Use integration by parts:

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

            Let $u{\left(u \right)} = \operatorname{acos}{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = 1$.

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

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

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

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

            Now evaluate the sub-integral.

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

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

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

               Then let $du = - 2 u du$ 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:

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

            So, the result is: $\sqrt{1 - u^{2}}$

         So, the result is: $\frac{u \operatorname{acos}{\left(u \right)}}{2} - \frac{\sqrt{1 - u^{2}}}{2}$

      Now substitute $u$ back in:

      $$\frac{x^{2} \operatorname{acos}{\left(x^{2} \right)}}{2} - \frac{\sqrt{1 - x^{4}}}{2}$$

   Method #2

   1. Use integration by parts:

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

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

      Then $\operatorname{du}{\left(x \right)} = - \frac{2 x}{\sqrt{1 - x^{4}}}$.

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

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

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

      Now evaluate the sub-integral.

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

      $$\int \left(- \frac{x^{3}}{\sqrt{1 - x^{4}}}\right)\, dx = - \int \frac{x^{3}}{\sqrt{1 - x^{4}}}\, dx$$

      1. Let $u = 1 - x^{4}$.

         Then let $du = - 4 x^{3} dx$ and substitute $- \frac{du}{4}$:

         $$\int \left(- \frac{1}{4 \sqrt{u}}\right)\, du$$

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

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

            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: $- \frac{\sqrt{u}}{2}$

         Now substitute $u$ back in:

         $$- \frac{\sqrt{1 - x^{4}}}{2}$$

      So, the result is: $\frac{\sqrt{1 - x^{4}}}{2}$

2. Add the constant of integration:

   $$\frac{x^{2} \operatorname{acos}{\left(x^{2} \right)}}{2} - \frac{\sqrt{1 - x^{4}}}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \operatorname{acos}{\left(x^{2} \right)}}{2} - \frac{\sqrt{1 - x^{4}}}{2}+ \mathrm{constant}$$