Question

Indefinite Integral using partial fraction decomposition method:

Here's what I'm supposed to integrate:
(2x+1)/(2x^2+4x)

and my work:
(2x+1)/(2x^2+4x)
(2x+1)/(2x(x+2))
(2x+1)/(2x(x+2))= (A/2x)+(B/(x+2))
(2x+1)/(2x(x+2))= ((A)(x+2)+(B(2x))/(2x(x+2)
(2x+1) = 2Ax +B(x+2)

this is where I'm stuck, as you can see it's the algebraic steps that I'm having trouble with more than anything. Thanks for the help!

Answer 1

instead of getting the A's and B's separated you need to get the x and the constant separated and then you just compare the coefficients and you will have 2 simultaneous equations with 2 unknowns

Answer 2

(2x+1)= x(2A+B) + (2B)
so
2A+B=2 and 2B=1

Answer 3

Thanks a lot for the prompt response!

I feel silly, as it's been a while since I've done any math using a system of equations (if that's the right word to use) but can you show me how (2x+1)=x(2A+B)+2B reduces down to 2A+B=2 and 2B=1

I also got B = 1/2 before using a guess and check method, and A = 3/4, which creates an easy-ish equation to integrate but it still wasn't the antiderivative of the opening problem so I felt I went wrong on this step. so if you could elaborate on what you wrote that would be a huge help. Thanks!!

Answer 4

lemme do it nicely for you quick one second

Answer 5

ok thank you adbermie. Take your time

Answer 6

\[\frac{ 2x+1 }{ 2x^2+4x }=\frac{ 2x +1}{ 2x(x+2) }\]
\[\frac{ 2x+1 }{ 2x(x+2 )}=\frac{ A }{ 2x }+\frac{ B }{ x+2 }\]
multiply by the denominator of the left hand side
\[2x+1=A(x+2)+B(2x)=(A+2B)x+2A\]
and now you just need to make sure the coefficient of x and the constant are the same on both sides. it seems you just made a strange arithmetic error in your working
so you end up with
\[2A+B=2 \] and \[2A=1\]
ie.
\[A=\frac{ 1 }{ 2 }\] and thus \[B=1\]

Answer 7

sorry haha i made a stupid mistake

Answer 8

should be A+2B=2

Answer 9

so B=3/4

Answer 10

what did the answers give you as the antiderivative?

Answer 11

hey well i appreciate you writing that all out for me adbermie, i understand how to get A & B now. the antiderivative i'm getting is 1/2ln|x+2| + 3/8ln|x|+C, but that doesnt derive to what we start with. once again thanks a lot for your help!

Answer 12

sorry for responding a little late my internet connection was spotty

Answer 13

did u get A=1/4, B=3/4 ?

Answer 14

nope, I got A =1/2, B=3/4

Answer 15

sorry A=1/2

Answer 16

yeah thats correct

Answer 17

now just put it in (2x+1)/(2x(x+2))= (A/2x)+(B/(x+2))

Answer 18

Try using integration by recognition. The derivative of the bottom term is almost on the top, which you can force with some alegbra

Answer 19

hey thanks for the responses guys.

hartnn- shouldnt A and B be swapped? so its A/x+2 + B/2x

thanks for the suggestion ZhangA. I'm not seeing how that would work though off the bat..

Answer 20

why swapping? NO
(2x+1)/(2x(x+2))= (A/2x)+(B/(x+2)) = 1/4x + 3/4(x+2)
and thats it, just integrate

Answer 21

haha, I wasnt swapping for the sake of swapping. Above I ended up with the terms A/x+2 + B/x+2...but the step before that was vice versa, like you have. let me see tho

Answer 22

alright cool! thanks hartnn. I'm getting the same answer as wolframalpha