# Partial Fraction Decomp. Integral

## Homework Statement

$$\int\frac{4y^2-7y-12}{y(y+2)(y-3)}$$ with limits of integration $$[1, 2]$$

## Homework Equations

The answer given in the book is $$\frac{27}{5}ln(2)-\frac{9}{5}ln(3)$$

or $$\frac{9}{5}ln\frac{8}{3}$$

## The Attempt at a Solution

$$\int\frac{4y^2-7y-12}{y(y+2)(y-3)}$$

$$=\int\frac{A}{y}+\frac{B}{y+2}+\frac{C}{y-3}$$

I found:

$$4y^2-7y-12=A(y+2)(y-3)+B(y)(y-3)+C(y)(y+2)$$

If y=-2, then $$B=1$$

I also found:

$$4y^2-7y-12=(A+B+C)y^2+(2C-3B-A)y-6A$$

so $$6A=12$$, or $$A=2$$.

Also, $$(A+B+C)=4$$ so $$(2+1+C)=4$$ and $$C=1$$

Therefore: $$=\int\frac{A}{y}+\frac{B}{y+2}+\frac{C}{y-3}$$

$$=\int\frac{2}{y}+\frac{1}{y+2}+\frac{1}{y-3}$$

$$=2ln|y|+ln|y+2|+ln|y-3|$$ with limits $$[1, 2]$$

$$=2(ln(2)-ln(1))+(ln(2+2)-ln(1+2)+(ln|2-3|-ln|1-3|)$$

$$=2ln(2)+(ln(4)-ln(3)+(ln(1)-ln(2))$$

$$=2ln(2)+ln(\frac{4}{3})-ln(2)$$

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?

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Mark44
Mentor
Check your work. I get A = 2, B = 9/5, and C = 1/5.

Dick