`int_(1)^(3) (dx)/(x^(2)(x+1))=(2)/(3)+log .(2)/(3)`

Explanation

To solve the integral \( \int_{1}^{3} \frac{dx}{x^2 (x+1)} \), we will first perform partial fraction decomposition on the integrand.

### Step 1: Partial Fraction Decomposition
We want to express \( \frac{1}{x^2 (x+1)} \) in the form:
\[ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} \]
Multiplying through by the denominator \( x^2 (x+1) \) gives:
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