To solve the integral of (9x + 2) divided by (x squared plus x minus 6), we'll use partial fractions. First, we need to factor the denominator. The quadratic x squared plus x minus 6 can be factored as (x plus 3) times (x minus 2). We can verify this by finding the roots of the quadratic equation, which are x equals negative 3 and x equals 2.
Now that we've factored the denominator, we can set up the partial fraction decomposition. We write the fraction as A over x plus 3, plus B over x minus 2. To find the values of A and B, we multiply both sides by the denominator, giving us 9x plus 2 equals A times x minus 2, plus B times x plus 3. To solve for the constants, we substitute special values of x. When x equals 2, the term with A vanishes, and we get 20 equals 5B, so B equals 4. When x equals negative 3, the term with B vanishes, and we get negative 25 equals negative 5A, so A equals 5.
Now that we've found A equals 5 and B equals 4, we can rewrite our integral as the integral of 5 over x plus 3, plus 4 over x minus 2. We can now integrate each term separately. For the first term, the integral of 5 over x plus 3 is 5 times the natural logarithm of the absolute value of x plus 3. For the second term, the integral of 4 over x minus 2 is 4 times the natural logarithm of the absolute value of x minus 2. The graphs show the two functions we're integrating, with their vertical asymptotes at x equals negative 3 and x equals 2.
Finally, we combine our results to get the complete solution. The integral of 9x plus 2 divided by x squared plus x minus 6 equals 5 times the natural logarithm of the absolute value of x plus 3, plus 4 times the natural logarithm of the absolute value of x minus 2, plus an arbitrary constant C. We can verify our answer by differentiating it. The derivative of 5 ln|x+3| is 5 over x+3, and the derivative of 4 ln|x-2| is 4 over x-2. Combining these fractions with a common denominator gives us 5 times x-2 plus 4 times x+3, all over (x+3)(x-2). This simplifies to 5x-10+4x+12 over (x+3)(x-2), which equals 9x+2 over (x+3)(x-2). This matches our original integrand, confirming our solution is correct.
Let's summarize what we've learned. To integrate rational functions like this one, we use partial fraction decomposition. The key steps are: First, factor the denominator completely. Then, set up the partial fractions and find the coefficients by substituting strategic values. Next, integrate each simpler fraction separately. Finally, combine the results with a constant of integration. For our problem, the final answer is 5 times the natural logarithm of the absolute value of x plus 3, plus 4 times the natural logarithm of the absolute value of x minus 2, plus an arbitrary constant C. This method works for any proper rational function where the denominator can be factored into linear or quadratic terms.