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1. \int \frac{2 x}{x^{2} + 1}\, dx

   Find an antiderivative.

2. \int \frac{2 x}{x^{2} + 1}\, dx = 2 \int \frac{x}{x^{2} + 1}\, dx

   Pull the constant 2 out of the integral.

3. \int \frac{x}{x^{2} + 1}\, dx = \frac{1}{2} \int \frac{2 x}{x^{2} + 1}\, dx

   Pull the constant \frac{1}{2} out of the integral.

4. u = x^{2} + 1,\quad du = 2 x\, dx

   Substitute u = x^{2} + 1.

5. \int \frac{x}{x^{2} + 1}\, dx = \int \frac{1}{u}\, d_u

   Rewrite the integral in terms of u.

6. \int \frac{1}{u}\, d_u = \log{\left(u \right)}

   ∫ 1/u du = ln|u|.

7. = \log{\left(x^{2} + 1 \right)}

   Substitute back u = x^{2} + 1.

8. F(x) = \log{\left(x^{2} + 1 \right)} + C

   Add the constant of integration.