Integral of (x^2+1)/x 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 $\frac{du}{2}$:

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

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

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

         1. Rewrite the integrand:

            $$\frac{u + 1}{u} = 1 + \frac{1}{u}$$

         2. Integrate term-by-term:

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

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

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

            The result is: $u + \log{\left(u \right)}$

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

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

      $$\frac{x^{2} + 1}{x} = x + \frac{1}{x}$$

   2. Integrate term-by-term:

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

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

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

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

2. Add the constant of integration:

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

The answer is:

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