# "Partial Fractions" Decomposition Integrals

• DameLight
In summary, the conversation is discussing the process of solving a integral problem with a denominator raised to a power. The speaker is breaking the fraction into two parts and assuming A / (x+7) for the first part, but is unsure of what to assume for the second part. With guidance, they realize that the numerator should be a polynomial one-degree lower than the denominator and make it more general by using Bx + C.
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.

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?

SteamKing said:
what should you assume?

Bx + B + 1?

DameLight said:
Bx + B + 1?

And why would you assume this?

SteamKing said:
And why would you assume this?

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

DameLight said:
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 : )

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

## 1. What is partial fractions decomposition in integration?

Partial fractions decomposition is a method used in integration to break down a complex rational expression into simpler fractions that can be easily integrated.

## 2. When is partial fractions decomposition used in integration?

Partial fractions decomposition is used when the integrand (the expression being integrated) is a rational function, meaning it is a ratio of two polynomials.

## 3. How do you perform partial fractions decomposition?

To perform partial fractions decomposition, the integrand is first factored into its irreducible factors. Then, the coefficients of each factor are determined by setting up a system of equations and solving for them. Finally, the partial fractions are combined and integrated individually.

## 4. What are the benefits of using partial fractions decomposition in integration?

Partial fractions decomposition simplifies the integration process by breaking down a complex rational expression into simpler fractions, making it easier to integrate. It also allows for the use of integration techniques such as substitution and integration by parts.

## 5. Are there any limitations to using partial fractions decomposition in integration?

Partial fractions decomposition can only be used when the integrand is a rational function. It is also limited to proper fractions, meaning the degree of the numerator must be less than the degree of the denominator. In some cases, partial fractions decomposition may result in complex fractions, which can be challenging to integrate.

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