Heaviside Method Solution to Problem 11.7-3

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

Even though using the Heaviside method is easier than the standard method, using a calculator is still recommended for this problem.

The problem is to find the indefinite integral

11x³ - 107x + 108 dx (x + 7)(x - 2)³

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. This denominator has repeated roots. Indeed the (x - 2) root is repeated three times. Recall how we set up repeated roots in the main text. Do this one that same way. Write that out, and then click here.

You should have

11x³ - 107x + 108 = (x + 7)(x - 2)³

A B + + x + 7 x - 2

C D + (x - 2)² (x - 2)³

although any ordering of A, B, C, and D over the four binomials listed would be ok.

Step 2: Go for the highest power first. That means solving first for D, which you do by multiplying the above through by (x - 2)³ and evaluating at x = 2. Then click here.

You should have

11x³ - 107x + 108 A(x - 2)³ = + (x + 7) x + 7

B(x - 2)² + C(x - 2) + D

= x = 2

                                                       88 - 214 + 108
                                                                       =  -2
                                                              9

Observe that you eliminate A, B, and C because their partial fractions are multiplied by zero. Hence you establish that D = -2 from this equation alone.

Step 3: Subtract out the solved partial fraction. You now know what D is, so you can subtract its partial fraction from the setup. You just have to put D's partial fraction over the common denominator by multiplying its numerator and denominator by (x + 7). Subtract that from both sides of the setup. Write the simplified equation, then click here.

You should have

     11x³  -  107x  +  108
   -(          -2x  -   14 )
                            
     11x³  -  105x  +  122

hence

11x³ - 105x + 122 = (x + 7)(x - 2)³

A B + + x + 7 x - 2

C (x - 2)²

Step 4: Divide out the common factor on the left. Since you have (x - 2)³ in the denominator on the left and only (x - 2)² as your maximum power in the denominators on the right, it must be true that (x - 2) divides evenly into the numerator on the left. So use polynomial long division to divide out the common factor. Then click here.

Your polynomial long division is

            11x²  + 22x   -   61         
   x - 2  ) 11x³  +  0x²  -  105x  +  122
            11x³  - 22x²
                        
                    22x²  -  105x
                    22x²  -   44x
                                 
                             -61x  +  122
                             -61x  +  122
                                         
                                        0

So your simplified equation is

11x² + 22x - 61 = (x + 7)(x - 2)²

A B + + x + 7 x - 2

C (x - 2)²

Step 5: Go for the highest power again. This time you will solve for C by doing that. Multiply the above equation through by (x - 2)² and evaluate it at x = 2. Then click here.

You should have

11x² + 22x - 61 = x + 7

A(x - 2)² + B(x - 2) + C x + 7

44 + 44 - 61 = = 3 x = 2 9

Observe that you eliminate A and B from the above because you've multiplied them by zero. Hence you establish that C = 3 from this equation alone.

Step 6: Subtract out the solved partial fraction from the simplified equation. You know what C is now. Put its partial fraction over the common denominator on the left side of the simplified equation. You will have to multiply numerator and denominator of the C partial fraction by (x + 7). Then subtract that partial fraction from both sides. When you've got it, click here.

You should have

     11x²  +  22x  -  61
   -(          3x  +  21 )
                          
     11x²  +  19x  -  82

hence

11x² + 19x - 82 = (x + 7)(x - 2)²

A B + x + 7 x - 2

Step 7: Divide out the common factor on the left. This time the denominator on the left has (x - 2)², whereas on the right the highest power of (x - 2) in any denominator is the first power. So it must be the case that the numerator on the left is evenly divisible by (x - 2). Use polynomial long division to divide that common factor out. Then click here.

Your polynomial long division is

             11x   +  41        
   x - 2  )  11x²  +  19x  -  82
             11x²  -  22x
                         
                      41x  -  82
                      41x  -  82
                                
                               0

So your simplified equation is

11x + 41 = (x + 7)(x - 2)

A B + x + 7 x - 2

Step 8: Solve for A by multiplying through by (x + 7) and evaluating at x = -7. We're down to a straight Heaviside problem with no repeated roots now, so it gets a lot easier. Do the above, then click here.

You should have

11x + 41 = (x - 2)

B(x + 7) A + x - 2

-77 + 41 = = 4 x = -7 -9

Observe that B is eliminated because you multiplied it by zero. Hence you have established that A = 4 from this equation alone.

Step 9: Solve for B by multiplying through by (x - 2) and evaluating at x = 2. You're getting very near the end now. Do this step, then click here.

You should have

11x + 41 = (x + 7)

A(x - 2) + B x + 7

22 + 41 = = 7 x = 2 9

Observe that A is eliminated because you multiplied it by zero. Hence you have established that B = 7 from this equation alone.

Step 10: Put in the known values for A, B, C, and D, and integrate. This is the last step. You have from previous steps: A = 4, B = 7, C = 3, and D = -2. Do the integration, then click here.

Here it is

11x³ - 107x + 108 dx = (x + 7)(x - 2)³

4 7 + + x + 7 x - 2

3 - (x - 2)²

2 (x - 2)³

dx =

3 4 ln|x + 7| + 7 ln|x - 2| - + x - 2

1 + K (x - 2)²

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