Integral of x^2/(1+x^3) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x^{3} + 1$.

      Then let $du = 3 x^{2} dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{1}{3 u}\, du$$

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

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

         1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

         So, the result is: $\frac{\log{\left(u \right)}}{3}$

      Now substitute $u$ back in:

      $$\frac{\log{\left(x^{3} + 1 \right)}}{3}$$

   Method #2

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

         $$\int \frac{2 x - 1}{3 \left(x^{2} - x + 1\right)}\, dx = \frac{\int \frac{2 x - 1}{x^{2} - x + 1}\, dx}{3}$$

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

            Then let $du = \left(2 x - 1\right) dx$ and substitute $du$:

            $$\int \frac{1}{u}\, du$$

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            Now substitute $u$ back in:

            $$\log{\left(x^{2} - x + 1 \right)}$$

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

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

         $$\int \frac{1}{3 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{3}$$

         1. Let $u = x + 1$.

            Then let $du = dx$ and substitute $du$:

            $$\int \frac{1}{u}\, du$$

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            Now substitute $u$ back in:

            $$\log{\left(x + 1 \right)}$$

         So, the result is: $\frac{\log{\left(x + 1 \right)}}{3}$

      The result is: $\frac{\log{\left(x + 1 \right)}}{3} + \frac{\log{\left(x^{2} - x + 1 \right)}}{3}$

2. Add the constant of integration:

   $$\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}$$