# Integration Help

• roam

#### roam

1. Homework Statement

Find the antiderivative:

$$\int\frac{2x^3-5x^2+5x-12}{(x-1)^2(x^2+4)}$$

## Homework Equations

3. The Attempt at a Solution
Using Integration by Partial Fractions:

$$\int\frac{2x^3-5x^2+5x-12}{(x-1)^2(x^2+4)} = \frac{A}{(x-1)}+\frac{B}{(x-1)^2}+\frac{Cx+D}{(x^2+4)}$$

2x3-5x2+5x-12 = A(x-1)(x2+4)+B(x2+4)+(Cx+D)(x-1)2

Multiplying out:

= -4A+4Ax-Ax2+Ax3+Bx2+4B+Cx+D-2Cx2-2Dx-Cx3+Dx2

Collecting like terms:

2x3-5x2+5x-12 = x3(A-C)+x2(B-2C+D-A)+x(4A+C-2D) + (D+4B)

Equating corresponding coefficients gives

A-C = 2
B-2C+D-A = -5
4A+C-2D =5
D+4B-4A=-12

in the matrix form:

1, 0, -1, 0, 2
-1, 1, -2, 1, -5
4, 0, 1, -2, 5
-4, 4, 0, 1, -12

I used MATLAB to get the reduced-row echelon form:

rref=

1, 0, 0, 0, 27
0, 1, 0, 0, 8
0, 0, 1, 0, 25
0, 0, 0, 1, 64

Therefore, A=27, B=8, C= 25 and D=64 (?)

$$\int \frac{27}{(x-1)}+ \int \frac{8}{(x-1)^2}+ \int \frac{25x+64}{(x^2+4)}$$

= 27 Log(-1 + x)-8/(x-1)+32arctan(x/2)+25/2log(4+x2)

I'm not sure if this is the correct answer to this problem, because I tried solving it using mathematica and I got:

$$\frac{2}{(x-1)} + log(x-1) + (1/2) log (4+x^2)$$

I really appreciate it if someone could show me my mistakes. Thanks!

Last edited:

$$2x^3-5x^2+5x-12 = A(x-1)(x^2+4)+B(x^2+4)+(Cx+D)(x-1)^2$$

is true for all values of x. Put x=1
you'd get -10=5B =>B=-2. So I think you did something wrong in multiplying out.

You got some of your coefficients wrong when you expanded the right side of your equation. After collecting like terms I got these coefficients:
x^3: A + C (you have A - C)
x^2: -A + B - 2C + D (same)
x: 4A + C - 2D (same)
1: -4A + 4B + D (you have 4B + D at first, but apparently added in the -4A term in later work)

I didn't work this through, but I did find that B = -2.

Thanks for your input guys! Changing "-1" to "1" made all the difference!

I row-reduced the new matrix and got:

A=1, B=-2, C=1 and D=0

And eventually I got the right answer. Thanks very much for spotting my error.