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**1. Homework Statement**

Find the antiderivative:

[tex]\int\frac{2x^3-5x^2+5x-12}{(x-1)^2(x^2+4)}[/tex]

Find the antiderivative:

[tex]\int\frac{2x^3-5x^2+5x-12}{(x-1)^2(x^2+4)}[/tex]

## Homework Equations

**3. The Attempt at a Solution**

Using Integration by Partial Fractions:

[tex]\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)}[/tex]

2x

Multiplying out:

= -4A+4Ax-Ax

Collecting like terms:

2x

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 (?)

[tex]\int \frac{27}{(x-1)}+ \int \frac{8}{(x-1)^2}+ \int \frac{25x+64}{(x^2+4)}[/tex]

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

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

[tex]\frac{2}{(x-1)} + log(x-1) + (1/2) log (4+x^2)[/tex]

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

Using Integration by Partial Fractions:

[tex]\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)}[/tex]

2x

^{3}-5x^{2}+5x-12 = A(x-1)(x^{2}+4)+B(x^{2}+4)+(Cx+D)(x-1)^{2}Multiplying out:

= -4A+4Ax-Ax

^{2}+Ax^{3}+Bx^{2}+4B+Cx+D-2Cx^{2}-2Dx-Cx^{3}+Dx^{2}Collecting like terms:

2x

^{3}-5x^{2}+5x-12 = x^{3}(A-C)+x^{2}(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 (?)

[tex]\int \frac{27}{(x-1)}+ \int \frac{8}{(x-1)^2}+ \int \frac{25x+64}{(x^2+4)}[/tex]

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

^{2})I'm not sure if this is the correct answer to this problem, because I tried solving it using mathematica and I got:

[tex]\frac{2}{(x-1)} + log(x-1) + (1/2) log (4+x^2)[/tex]

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

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