When I was doing one of the practice tests, I got stumped in the Partial Fractions question, just because of one calculation error. Here’s the problem I have been working on:
Evaluate the integral of: (-8x + 12) ÷ (x2 (x-3)) dx
I have no problem doing this until I realized that I needed to find both values (A and C). When I tried to do 3×3 systems of equations, I ended up getting fractional answers which was [latex]A=-3\frac{5}{7}[/latex], and [latex]C=-2\frac{2}{7}[/latex]. I felt very suspicious at this point.
Since I knew that I won’t get anywhere with the numbers they gave me, I started over again. I reset my partial fraction decomposition, but this time, I was careful. While redoing this problem, I realized that I have to look for zeros (using the zero-product rule), which is 0 and 3. In addition, I am more careful with my arithmetic (as I got B = 4, C = -4), so I could seamlessly continue solving for A. Then, I did x-substitution for -1, and then substituted B and C for appropriate values. Therefore, A = 4. Since the partial fraction decomposition is complete, all I had to do is to integrate. That’s it!
During the final exam (and for all in mathematics), you must check not only the procedure, but also the arithmetic you did. Because even one small miscalculation on your arithmetic can prevent you from completing the problem. Good luck on your exams!
Good catch, Anthony! Don’t worry if you encounter fractions on the final exam; some questions are carefully cooked up to have integer values for A, B, and C, but it’s actually a lot more common for them to be fractions. Good luck!