1 X^2+x Partial Fraction

[Elenio Guardado]

Inorder to use partial fractions, the degree of the numerator must be less than the degree of the denominator. If the numerator and denominator have the same degree, or the numerator is higher than the denominator, then you must use long division to find that.

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Previously on adding/subtracting rational expressions, we want to combine two or more rational expressions into a single fraction just like the example below. However, partial fraction decomposition (also known as partial fraction expansion) is precisely the reverse process of that. The following is an illustrative diagram to show the main concept.

This problem is easy, so think of this as an introductory example. I will start by factoring the denominator (take out [latex]x[/latex] from the binomial). Next, I will set up the decomposition process by placing [latex]A[/latex] and [latex]B[/latex] for each of the unique or distinct linear factors. The subsequent steps then involve getting rid of all the denominators by multiplying the [latex]LCD[/latex] (which is just the original denominator of the problem) throughout the entire equation.

I should end up with a simple equation where I can easily compare the coefficients of similar terms from both sides of the equation. As a result, I will get a system of linear equations with variables [latex]A[/latex] and [latex]B[/latex] that can be solved by either the Substitution Method or Elimination Method, whichever I prefer.

Example 1: Find $\displaystyle\int\frac6x^2-1\,dx$.

DO: Notice that substitution doesn't work here, but the trig substitution using secant will, and partial fractions will. Try the problem both ways to see which you prefer. The solution below uses partial fractions. Don't read forward until you have tried the problem.

Integration by partial fractions is an integration technique which uses partial fraction decomposition to simplify the integrand. The integrand is written as partial fractions and then evaluated using standard methods.

\beginalign \dfrac(x-1)^2(x+1)(x-1)^2(x+1) &= \dfracA(x-1)^2(x+1)x-1 \\ &\quad +\dfracB(x-1)^2(x+1)(x-1)^2 \\ &\quad+\dfracC(x-1)^2(x+1)x+1\\ 1 &= A(x-1)(x+1)+B(x+1)+C(x-1)^2. \endalign

This process of taking a rational expression and decomposing it into simpler rational expressions that we can add or subtract to get the original rational expression is called partial fraction decomposition. Many integrals involving rational expressions can be done if we first do partial fractions on the integrand.

Note that in most problems we will go straight from the general form of the decomposition to this step and not bother with actually adding the terms back up. The only point to adding the terms is to get the numerator and we can get that without actually writing down the results of the addition.

Before moving onto the next example a couple of quick notes are in order here. First, many of the integrals in partial fractions problems come down to the type of integral seen above. Make sure that you can do those integrals.

The next step is to set numerators equal. If you need to actually add the right side together to get the numerator for that side then you should do so, however, it will definitely make the problem quicker if you can do the addition in your head to get,

Again, as noted above, integrals that generate natural logarithms are very common in these problems so make sure you can do them. Also, you were able to correctly do the last integral right? The coefficient of \(\frac56\) is correct. Make sure that you do the substitution required for the term properly.

Note that we used \(x^0\) to represent the constants. Also note that these systems can often be quite large and have a fair amount of work involved in solving them. The best way to deal with these is to use some form of computer aided solving techniques.

Now, there is a variation of the method we used in the first couple of examples that will work here. There are a couple of values of \(x\) that will allow us to quickly get two of the three constants, but there is no value of \(x\) that will just hand us the third.

In the previous example there were actually two different ways of dealing with the \(x^2\) in the denominator. One is to treat it as a quadratic which would give the following term in the decomposition

We used the second way of thinking about it in our example. Notice however that the two will give identical partial fraction decompositions. So, why talk about this? Simple. This will work for \(x^2\), but what about \(x^3\) or \(x^4\)? In these cases, we really will need to use the second way of thinking about these kinds of terms.

Partial Fractions are the fractions that are formed when a complex rational expression is split into two or more simpler fractions. Generally, fractions with algebraic expressions are difficult to solve and hence we use the concepts of partial fractions to split the fractions into numerous subfractions. While decomposition, generally, the denominator is an algebraic expression, and this expression is factorized to facilitate the process of generating partial fractions. A partial fraction is a reverse of the process of the addition of rational expressions.

In the normal process, we perform arithmetic operations across algebraic fractions to obtain a single rational expression. This rational expression, on splitting in the reverse direction involved the process of decomposition of partial fractions and results in the two partial fractions. Let us learn more about partial fractions in the following sections.

When a rational expression is split into the sum of two or more rational expressions, the rational expressions that are a part of the sum are called partial fractions. This is referred to as splitting the given algebraic fraction into partial fractions. The denominator of the given algebraic expression has to be factorized to obtain the set of partial fractions.

In the above example, the numerators of partial fractions are 1 and 3. The numerator of a partial fraction is not always a constant. If the denominator is a linear function, the numerator is constant. And, if the denominator is a quadratic equation, then the numerator is linear. It means, the numerator's degree of a partial fraction is always one less than the denominator's degree. Further, the rational expression needs to be a proper fraction to be decomposed into a partial fraction. Listed below in the table are partial fraction formulas (here, all variables apart from x are constants).

The partial fraction decomposition is writing a rational expression as the sum of two or more partial fractions. The following steps are helpful to understand the process to decompose a fraction into partial fractions.

When we have to decompose an improper fraction into partial fractions, we first should do the long division. The long division is helpful to give a whole number and a proper fraction. The whole number is the quotient in the long division, and the remainder forms the numerator of the proper fraction, and the denominator is the divisor. The format of the result of the long division would be Quotient + Remainder/Divisor. Let us understand more of this with the help of the below example.

The partial fraction is the result of writing a rational expression as the sum of two or more fractions. First simplify the rational expression by breaking it down into the possible factors for the numerator, and the denominator. Further, split the expression into partial fractions based on the formulas. The formulas for partial fractions depend on the number of factors and the degree of the denominator of the rational expression. Further, find the value of the required constants to solve the partial fractions.

The word "partial" means a "part" and hence a partial fraction is one of the fractions when a given fraction is decomposed into the sum of multiple fractions. The input for the process of the partial fractions is a rational expression, and the result is the sum of two or more proper fractions.

The partial fraction decomposition is to be used when the denominator of the fraction is an algebraic expression, and when there is a need to split the fraction. Also, there should be a possibility of getting at least two factors for the algebraic expression in the denominator.

The types of partial fractions depend on the number of possible factors of the denominator, and the degree of the factors of the denominator. Broadly there are about three types of partial fractions. The following three types of partial fractions are as follows.

For the process of getting partial fractions, the given fraction needs to be a proper fraction. If the given fraction is an improper fraction, the numerator is divided by the denominator to obtain a quotient and a remainder. And the fraction that is used to split into partial fractions in this case would be the remainder/denominator.

The word \"partial\" means a \"part\" and hence a partial fraction is one of the fractions when a given fraction is decomposed into the sum of multiple fractions. The input for the process of the partial fractions is a rational expression, and the result is the sum of two or more proper fractions.

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