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How would i do the following indefinite integration $$\int \frac{1}{x^2+10x+21} \, dx$$

so far I've turned the bottom polynomial into $(x+7)(x+3)$ not too sure where to go from here

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1 Answer 1

up vote 3 down vote accepted

Hint: Partial Fractions: $$\frac{1}{(x+7)(x+3)}=\frac{A}{x+7}+\frac{B}{x+3}$$ Now find $A,B$ and use the linearity of the integral: $$\int \frac{1}{(x+7)(x+3)} dx=\int \frac{A}{x+7}dx+\int \frac{B}{x+3}dx$$ This should be simple now.

EDIT: Evaluating $A,B$: $$\frac{1}{(x+7)(x+3)}=\frac{A}{x+7}+\frac{B}{x+3}\iff 1=A(x+3)+B(x+7)$$ Setting $x=-3$ and $x=-7$ give $A,B=?$

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oh dear i cant believe i didnt think of this!! thankyou :) –  jill Dec 22 '12 at 11:53
    
urmm one quick question from the above partial integrals how would i work out the values of A and B? –  jill Dec 22 '12 at 12:02
    
i understand the answer would be $Alog(x+7) + Blog(x+3)$ but dont really know how to work out A and B –  jill Dec 22 '12 at 12:04
    
$A= -1/4$ and $B=1/4$ thankyou :D –  jill Dec 22 '12 at 12:06
    
@jill Exactly!! –  Nameless Dec 22 '12 at 12:07

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