The given problem is to evaluate the definite integral:

$\int_{1}^{2} \left(\frac{4}{x^2} + 1\right) \, dx.$

Solution:

The integral can be split into two simpler integrals:

$\int_{1}^{2} \left(\frac{4}{x^2} + 1\right) \, dx = \int_{1}^{2} \frac{4}{x^2} \, dx + \int_{1}^{2} 1 \, dx.$

1. Solve $\int_{1}^{2} \frac{4}{x^2} \, dx$:

Recall that $\frac{1}{x^2}$ can be rewritten as $x^{-2}$, and the integral of $x^{-n}$ is:

[ \int x^{-n} , dx = \frac{x^{-n+1}}{-n+1} + C \quad \