Standard Method Solution to Problem 11.7-1 © 2001 by Karl Hahn

This page applies the standard method of partial fractions to find the solution. To see the Heaviside method applied to this same problem, click here.

The problem is to find the indefinite integral

-6x3 + 71x2 + 283x - 1614 dx (x - 4)(x - 3)(x + 5)(x + 6)

Step 1: Set it up. The integrand is a rational function with a fourth degree polynomial in its denominator. So there will be four unknowns. You can call them, A, B, C, and D, although any four distinct symbols would have been ok. There are no repeated roots in the denominator, and all the roots are real. So to set it up, just put one of the unknowns over each binomial and equate the sum of those partial fractions to the integrand. Write that out, and then click here.

Step 2: Put all the partial fractions over the common denominator and then arrange the terms according to powers of x. You will have to multiply out each of the following:
 (x - 3)(x + 5)(x + 6), 
 (x - 4)(x + 5)(x + 6), 
 (x - 4)(x - 3)(x + 6),  and
 (x - 4)(x - 3)(x + 5). 
Each is multiplied by one of the unknowns (you have to figure out which goes with which), and the resulting products added together to form the numerator on the right. Work it through, then click here.

Step 3: Equate the numerators according to powers of x. This will give you four linear equations in A, B, C, and D. Go ahead and write them all out, then click here.

Step 4: Solve the linear equation. This is not as bad as you think. Each step is simple. It's just that there are a lot of steps and it is easy to make arithmetic and sign errors. Try doing the Gaussian elimination process yourself. Then click here to see if you got the same results I did. Using a calculator helps a lot, but you still have to keep careful track of all your signs.

Here is the procedure for solving the linear equations in the example for A, B, C, and D by the method of Gaussian elimination. Each procedure tells you how to get from the current set of equations to the next set of equations. (Or you can go to QuickMath, enter in all four equations and the list of variables, and have it do it all for you, but what fun is that?)

Equations		Procedure
A + B + C + D = -6 8A + 7B - C - 2D = 71 -3A - 14B - 30C - 23D = 283 -90A - 120B + 72C + 60D = -1614		Subtract 8 times first equation from second equation. Add 3 times first equation to third equation. Add 90 times first equation to fourth equation.
A + B + C + D = -6 -B - 9C - 10D = 119 -11B - 27C - 20D = 265 -30B + 162C + 150D = -2154		Subtract 11 times second equation from third equation. Subtract 30 times second equation from fourth equation.
A + B + C + D = -6 -B - 9C - 10D = 119 72C + 90D = -1044 432C + 450D = -5724		Subtract 6 times third equation from fourth equation. Then divide fourth equation by -90
A + B + C + D = -6 -B - 9C - 10D = 119 72C + 90D = -1044 D = -6		Subtract 90 times fourth equation from third equation. Then divide fourth equation by 72.
A + B + C + D = -6 -B - 9C - 10D = 119 C = -7 D = -6		Add 10 times fourth equation to second equation. Add 9 times third equation to second equation. Then multiply the resulting second equation by -1.
A + B + C + D = -6 B = 4 C = -7 D = -6		Subtract the second, third, and fourth equations all from the first.
A = 3 B = 4 C = -7 D = -6		Done.

Step 5: Put the solved coefficients in and integrate. Go back to your expansion into partial fractions (step 1) and substitute A, B, C, and D with the solutions you just got. Then integrate the result. That last part is pretty easy. Then click here.

              3 ln|x - 4|  +  4 ln|x - 3|  -  7 ln|x + 5|  -  6 ln|x + 6|  +  K

Return to Main Text

You can email me by clicking this button: