# Integrate Partial Fractions: x^3/(x^2-1)

• Firepanda
In summary, the conversation discusses the use of partial fractions to integrate a given equation. It is pointed out that the degree of the numerator must be less than the degree of the denominator for partial fractions to work. However, the given equation does not have this property. The conversation then moves on to discuss using a substitution method to evaluate a definite integral, with the suggested substitution being t=lnx.
Firepanda
Integrate using partial fractions:

(int) (x^3)/(x^2 -1) dx

I have put into the form (int) (x^3)/((x-1)(x+1)) dx

I thought partial fractions had this property:

'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'

And this obviously doesn't, so how do you do it?

Firepanda said:
Integrate using partial fractions:

(int) (x^3)/(x^2 -1) dx

I have put into the form (int) (x^3)/((x-1)(x+1)) dx

I thought partial fractions had this property:

'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'

And this obviously doesn't, so how do you do it?

Divide! What is x3 divided by x2- 1? What are the quotient and remainder?

ahhh

so it =

x + 1/(x-1) - 1/(x+1)(x-1)

:D

While I am here I have another question:

(int) 1 to e (ln(x))/x^2 dx

THe question says, use a suitable substitution to evaluate the definite integral, I can do it by parts but I don't want to lose marks for it.

Can any1 suggest the substitution to get me started?
Thx

Firepanda said:
While I am here I have another question:

(int) 1 to e (ln(x))/x^2 dx

THe question says, use a suitable substitution to evaluate the definite integral, I can do it by parts but I don't want to lose marks for it.

Can any1 suggest the substitution to get me started?
Thx
This should be in the Calculus section.

Verifying your problem ...

$$\int_1^e\frac{\ln x}{x^2}dx$$

rocophysics said:
This should be in the Calculus section.

Verifying your problem ...

$$\int_1^e\frac{\ln x}{x^2}dx$$

Yep that's the one

Firepanda said:
Yep that's the one
Substitutions ...

$$t=\ln x \rightarrow e^t=x$$

$$dt=\frac 1 x dx$$

You will need to do parts afterwards.

## 1. How do I determine the partial fraction decomposition of a given fraction?

The first step in determining the partial fraction decomposition of a fraction is to factor the denominator into linear and irreducible quadratic factors. Then, you can use the method of undetermined coefficients to find the constants that will make up the partial fraction decomposition.

## 2. Can any fraction be decomposed into partial fractions?

Yes, any proper rational function (a fraction where the degree of the numerator is less than the degree of the denominator) can be broken down into partial fractions.

## 3. What is the purpose of integrating partial fractions?

Integrating partial fractions allows us to break down a complex function into simpler, easier-to-integrate parts. This can be especially useful when dealing with integrals that involve fractions with higher degree polynomials in the numerator and denominator.

## 4. What are the different types of partial fractions?

There are three types of partial fractions: proper, improper, and mixed. A proper partial fraction has a degree of the numerator that is less than the degree of the denominator. An improper partial fraction has a degree of the numerator that is greater than or equal to the degree of the denominator. A mixed partial fraction has both a whole number and a proper fraction as its terms.

## 5. Are there any shortcuts or tricks for integrating partial fractions?

There are a few shortcuts that can be used in certain cases, such as when the denominator has repeated linear factors or when the fraction has a quadratic denominator with a linear numerator. However, these shortcuts are not applicable in all cases and it is important to understand the general method for integrating partial fractions.

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