### Integral Calculation In this section, we will tackle the problem of evaluating the following integral: \[ \int \frac{x}{x^2 + x - 2} \, dx \] This integral appears to be a rational function where the numerator has a lower degree than the polynomial in the denominator. To approach this, we can start by factoring the denominator to simplify the integrand, possibly using partial fraction decomposition. **Steps to Solve the Integral:** 1. **Factor the denominator**: The polynomial \( x^2 + x - 2 \) can be factored into \((x+2)(x-1)\). \[ x^2 + x - 2 = (x+2)(x-1) \] 2. **Rewrite the integral**: Substitute the factored expression into the integrand: \[ \int \frac{x}{(x+2)(x-1)} \, dx \] 3. **Partial Fraction Decomposition**: We can use partial fractions to decompose \( \frac{x}{(x+2)(x-1)} \) into simpler fractions that are easier to integrate. \[ \frac{x}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1} \] To find the constants \(A\) and \(B\), we will solve: \[ x = A(x-1) + B(x+2) \] By solving the system of equations, we determine the values of \(A\) and \(B\). 4. **Integrate each term**: Integrate the resulting partial fractions separately. 5. **Combine the results**: Sum the integrals of the partial fractions to find the final answer. Through these steps, students can systematically approach and solve the integral \( \int \frac{x}{x^2 + x - 2} \, dx \).