Integral of xdx/x^2+1 dx

1. Integrate term-by-term:

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

      $$\int \frac{1}{x}\, dx = \frac{\int \frac{2}{x}\, dx}{2}$$

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

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

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

         1. Don't know the steps in finding this integral.

            But the integral is

            $$\log{\left(u \right)}$$

         Now substitute $u$ back in:

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

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

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

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

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

2. Add the constant of integration:

   $$x + \frac{\log{\left(x^{2} \right)}}{2}+ \mathrm{constant}$$

The answer is:

$$x + \frac{\log{\left(x^{2} \right)}}{2}+ \mathrm{constant}$$