To integrate the expression $\frac{1+x}{x^2+1}$, we can use the substitution $u = x^2 + 1$.

First, we differentiate $u$ with respect to $x$:

which implies that

Now, we can substitute into the integral:

This expression can be split into two separate integrals:

The first integral, $\int \frac{1}{u} \, du$, evaluates to $\ln|u|$, and the second integral, $\int \frac{x}{u} \, du$, requires us to express $x$ in terms of $u$. From our substitution, we have:

Thus, we can express the second integral as:

However, for simplicity, let’s go back to our original integral. After performing the integrations, we will combine the results:

1. The first part gives us:

2. The second part (which we will evaluate later or note that it simplifies in the process) will contribute to the total integral.

Combining these results, we arrive at:

Ultimately, the integral simplifies to:

where $C$ is the constant of integration.

In conclusion, the integral of $\frac{1+x}{x^2+1}$ is: