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Am I integrating this right: (x^2 + 3x + 11)/(x+2)^4 ?

  • Thread starter Lo.Lee.Ta.
  • Start date
Am I integrating this right: (x^2 + 3x + 11)/(x+2)^4 ???

1. ∫(x2 + 3x + 4)/(x+2)4dx


2. With these sorts of problems, I think about integration by partial fractions.

So in this case, the denominator factors are all the same, so I have to make each one with different exponents.
I wrote:

A/(x + 2) + B/(x + 2)2 + C/(x + 2)3 + D/(x + 2)4

I multiplied to get common denominators:

A(x + 2)3 + B(x + 2)2 + C(x + 2) + D = x2 + 3x + 11

Now I need to figure out what A, B, C, and D equal.

When I substitute x=-2, D=9

A(x + 2)3 + B(x + 2)2 + C(x + 2) + 9 = x3 + 3x + 11

I think I am supposed to take the derivative now to figure out the other values.

2x + 3 = 3A(x + 2)2 + 2B(x + 2) + C

When I substitute x=-2, C= -1

Now, am I able to take the derivative again?
That's what I did next:
Substituting x=-2,

2 = 6A(x + 2) + 2B
B=1

If x=1,

2 = 6A(x + 2) + 2(1)
A=0

...Am I even doing this right? :/
I don't want to go on if I'm not even doing this part right!
Thank you very much! :D
 

Dick

Science Advisor
Homework Helper
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1. ∫(x2 + 3x + 4)/(x+2)4dx


2. With these sorts of problems, I think about integration by partial fractions.

So in this case, the denominator factors are all the same, so I have to make each one with different exponents.
I wrote:

A/(x + 2) + B/(x + 2)2 + C/(x + 2)3 + D/(x + 2)4

I multiplied to get common denominators:

A(x + 2)3 + B(x + 2)2 + C(x + 2) + D = x2 + 3x + 11

Now I need to figure out what A, B, C, and D equal.

When I substitute x=-2, D=9

A(x + 2)3 + B(x + 2)2 + C(x + 2) + 9 = x3 + 3x + 11

I think I am supposed to take the derivative now to figure out the other values.

2x + 3 = 3A(x + 2)2 + 2B(x + 2) + C

When I substitute x=-2, C= -1

Now, am I able to take the derivative again?
That's what I did next:
Substituting x=-2,

2 = 6A(x + 2) + 2B
B=1

If x=1,

2 = 6A(x + 2) + 2(1)
A=0

...Am I even doing this right? :/
I don't want to go on if I'm not even doing this part right!
Thank you very much! :D
If your original integrand was (x^2+3x+11)/(x+2)^4 (you probably have a typo), then, yes, you got the partial fractions right.
 
Oh, yes, was a typo! Thanks for letting me know it's right so far! :)
 
Alright, now I have:

∫(0/x+2) + (1/(x+2)2) - 1/(x+2)3 + 9/(x+2)4dx

U-substitution!

u= x+2
du=dx

∫u-2 - u-3 + 9u-4du

= -u-1 - u-2/-2 + 9u-3/-3

= -1/(x+2) + 1/(2(x+2)2) - 2/(x+2)-3

...Is this right?
Thanks so much! :)
 

SammyS

Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,095
881
Alright, now I have:

∫(0/(x+2)) + (1/(x+2)2) - 1/(x+2)3 + 9/(x+2)4dx

U-substitution!

u= x+2
du=dx

∫u-2 - u-3 + 9u-4du

= -u-1 - u-2/-2 + 9u-3/-3

= -1/(x+2) + 1/(2(x+2)2) - 2/(x+2)-3

...Is this right?
Thanks so much! :)
You can check by taking the derivative of the result.

By the way: This problem can be done more easily by using u-substitution at the outset, with u = x+2 .
 

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