partial fractions: (6x+7)/(x+2)^2

thebenji

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Sep 2, 2006
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I have to solve the indefinite integral of:

(6x+7)/(x+2)^2

So I split it:

(6x+7)/(x+2)^2 = A/(x+2) + B/(x+2)^2

Therefore

6x+7 = A(x+2)^2 + B(x+2)

How do I find A and B? Am I even on the right track? Thanks.
 
(6x+7)/(x+2)^2 = A/(x+2) + B/(x+2)^2

correction ...

6x+7 = A(x+2) + B

6x+7 = Ax + 2A + B

A has to = 6

2A + B has to equal 7 ... B = -5
 
Thank you, I can see why the B does not need an (x+2) and the A only needs an (x+2) to the first power.
 
Hello, thebenji!

I have to solve: \(\displaystyle \L\:\int\frac{6x\,+\,7}{(x\,+\,2)^2}\,dx\)

So I split it: 6x+7(x+2)2=Ax+2+B(x+2)2\displaystyle \:\frac{6x\,+\,7}{(x\,+\,2)^2} \:=\:\frac{A}{x\,+\,2}\,+\,\frac{B}{(x\,+\,2)^2}

Therefore: 6x+7=A(x+2)2+B(x+2)\displaystyle \:6x\,+\,7 \:= \:A(x\,+\,2)^2\,+\,B(x\,+\,2) . . . no

How do I find A\displaystyle A and B\displaystyle B?

You had: \(\displaystyle \L\:\frac{6x\,+\,7}{(x\,+\,2)^2} \:=\:\frac{A}{x\,+\,2}\,+\,\frac{B}{(x\,+\,2)^2}\)


Multiply through by the LCD, (x+2)2\displaystyle (x+2)^2

. . and we have: \(\displaystyle \L\:6x\,+\,7\;=\;A(x\,+\,2)\,+\,B\)



Now substitute some "good" values for x\displaystyle x.

Let \(\displaystyle x\,=\,-2:\;\;6(-2)\,+\,7\;=\;A(-2\,+\,2) \,+\,B\;\;\Rightarrow\;\;\L B\,=\,-5\)

Let x=0:    6(0)+7  =  A(0+2)+B        7=2A+B\displaystyle x\,=\,0:\;\;6(0)\,+\,7\;=\;A(0\,+\,2)\,+\,B\;\;\Rightarrow\;\;7 \:=\:2A\,+\,B
. . . . . . . . . Since B=5\displaystyle B = -5, we have: \(\displaystyle \,7 \:=\:2A\,-\,5\;\;\Rightarrow\;\;\L A\,=\,6\)


Therefore: \(\displaystyle \L\:\frac{6x\,+\,7}{(x\,+\,2)^2} \;= \;\frac{6}{x\,+\,2}\,-\,\frac{5}{(x\,+\,2)^2}\)

 
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