"Partial Fractions" Decomposition Integrals

[DameLight]

Hello,

I was just introduced to this concept and I have solved a few problems, but I haven't come across any with denominators to a raised power yet.

∫ 1 / [(x+7)(x^2+4)] dx

I would appreciate any directed help.

1. from the initial state I have broken the fraction into two assuming that (x+7)(x^2+4) is the common denominator where A and B are unknown.

∫ A / (x+7) + B / (x^2+4)

B / (x^2+4) has me confused

as I said before, I have not come across denominators to a raised power before and understand that the numerator needs to be raised to one power less, but what this looks like I don't know.

 

[SteamKing]

The first part of your PF decomp, A / (x + 7), is OK. For the second part, B / (x² + 4), you should assume the numerator is a polynomial one-degree lower than the denominator, which is why you assume A / (x + 7).

For the second part, instead of just B in the numerator, what should you assume?

 

[DameLight]

what should you assume?

Bx + B + 1?

 

[SteamKing]

Bx + B + 1?

And why would you assume this?

 

[DameLight]

And why would you assume this?

because the polynomial on the bottom is in degrees of x so B ( x + 1 ) is better?

 

[SteamKing]

because the polynomial on the bottom is in degrees of x so B ( x + 1 ) is better?

You're assuming that the coefficient of x and the constant will be the same. That's a bad assumption.
Make the numerator the more general Bx + C to cover all possibilities.

 

[DameLight]

ah I see now thank you for your help : )

 

[HallsofIvy]

"In degrees of x"? Surely you meant to say "second degree in x"!