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Homework Statement
[tex]\int\frac{4y^2-7y-12}{y(y+2)(y-3)}[/tex] with limits of integration [tex][1, 2][/tex]
Homework Equations
The answer given in the book is [tex]\frac{27}{5}ln(2)-\frac{9}{5}ln(3)[/tex]
or [tex]\frac{9}{5}ln\frac{8}{3}[/tex]
The Attempt at a Solution
[tex]\int\frac{4y^2-7y-12}{y(y+2)(y-3)}[/tex]
[tex]=\int\frac{A}{y}+\frac{B}{y+2}+\frac{C}{y-3}[/tex]
I found:
[tex]4y^2-7y-12=A(y+2)(y-3)+B(y)(y-3)+C(y)(y+2)[/tex]
If y=-2, then [tex]B=1[/tex]
I also found:
[tex]4y^2-7y-12=(A+B+C)y^2+(2C-3B-A)y-6A[/tex]
so [tex]6A=12[/tex], or [tex]A=2[/tex].
Also, [tex](A+B+C)=4[/tex] so [tex](2+1+C)=4[/tex] and [tex]C=1[/tex]
Therefore: [tex]=\int\frac{A}{y}+\frac{B}{y+2}+\frac{C}{y-3}[/tex]
[tex]=\int\frac{2}{y}+\frac{1}{y+2}+\frac{1}{y-3}[/tex]
[tex]=2ln|y|+ln|y+2|+ln|y-3|[/tex] with limits [tex][1, 2][/tex]
[tex]=2(ln(2)-ln(1))+(ln(2+2)-ln(1+2)+(ln|2-3|-ln|1-3|)[/tex]
[tex]=2ln(2)+(ln(4)-ln(3)+(ln(1)-ln(2))[/tex]
[tex]=2ln(2)+ln(\frac{4}{3})-ln(2)[/tex]
This is not equal to either answer provided in the book but I can't find an error. Also, how do you put the limits of integration into the function using Latex?