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Partial Fraction Decomposition

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In mathematics, the techniques of partial fractions are utilized at times in order to integrate and even in various other problems. These techniques are applied on rational fractions. A rational fraction is referred to a fraction having polynomials in numerator and denominator.

Partial fraction decomposition:
is the process of expressing a rational fraction in terms of the sum of a polynomial and one or more fractions with a more simple denominator. Partial fraction decomposition is an important concept since it is very useful in the computation of anti derivative or integration of a rational number. We can also define partial fraction decomposition as:

$\frac{P(x)}{Q(x)}$ = $\sum_{i}$ $\frac{P_{i}(x)}{Q_{i}(x)}$
Where, P(x) and Q(x) are polynomials in x.


The above mathematical relation explains symbolically that the rational fraction $\frac{P(x)}{Q(x)}$ is expressed in the form of sum of other simpler or lower degree rational expressions. The degree of $P_{i}(x)$ and $Q_{i}(x)$ are  usually lower than that of P(x) and Q(x).

We may also define partial fraction decomposition as the reverse procedure of a more elementary addition operation on rational fraction which results in a single rational fraction having higher degree numerator and denominator. The output of a full partial fraction decomposition expresses the given fraction as a sum of other fractions.

One point that is to be remembers is:
Partial fraction decomposition can only be done when the degree of numerator is strictly less than that of denominator; i.e in above relation, the degree of P(x) must be less than the degree of Q(x).

Rules

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There are few rules that are to be following while decomposing a ration fraction by partial fraction technique. These rules are illustrated below:

1) First, the rational fraction is to be observed for the suitable format for partial fractions. Examine the degree of polynomials in numerator and denominator. If the degree of numerator is greater than or equal to that of denominator, we need to divide the numerator by denominator before finding partial fractions.

For Example:

$\frac{x^{2}}{x+1}$ =(x-1)+ $\frac{1}{x+1}$ 

Now, we shall find partial fraction of the rational fraction $\frac{1}{x+1}$

2) There are various methods of finding partial fraction according different formats of given rational fractions. Students should observe them closely and should apply a suitable technique to them.

3) Before finding partial fraction, one needs to factorize the denominator as simplest as possible.

Table or Formulae

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The methods of partial fraction decomposition are to be chosen on the basis of the simplest form of rational expression in the denominator. The table describing various formulae and examples are as follows :
Type of factor in the denominator
Partial fraction technique
Example 
 (p x + q) $\frac{A}{p x + q}$
$\frac{3}{(2x+1)(x+2)}$ = $\frac{A}{2x+1}$+$\frac{B}{x+2}$
(p x + q)$^{n}$
$\frac{A_{1}}{p x + q}$ +$\frac{A_{2}}{(p x + q)^{2}}$ +...+$\frac{A_{n}}{(p x + q)^{n}}$
$\frac{4x}{(x-3)^{2}}$ = $\frac{A}{(x-3)}$ + $\frac{B}{(x-3)^{2}}$
px$^{2}$ + q x + r
$\frac{Ax+B}{px^{2} + q x + r}$
$\frac{2x}{3x^{2}-5}$ = $\frac{Ax+B}{3x^{2}-5}$
$(p x^{2} + qx + r)^{n}$   $\frac{A_{1}x+B_{1}}{p x^{2}+qx+r}$ + $\frac{A_{2}x+B_{2}}{(px^{2}+qx+r)^{2}}$ +...+$\frac{A_{n}x+B_{n}}{(p x^{2} + qx+r)^{n}}$
$\frac{3x}{(x^{2}-5)^{2}}$ = $\frac{Ax+B}{x^{2}-5}$+$\frac{Cx+D}{(x^{2}-5)^{2}}$

How to Find

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In order to find partial fractions of a rational fraction, we should follow the step given below:

Step 1: Check whether the degree of numerator is less than that of denominator. If it is not, then perform the division operation so that the degree of numerator of rational fraction becomes less than its denominator.

Step 2: In order to start partial fractions of such rational fraction, identify the exact format and apply the appropriate formula.

Step 3: Find the values of variables (A, B, C, ...) introduced through the suitable formula.

Step 4: Substitute the values of these variables in the equation obtained by applying the formula. In this way, we get the required partial fraction decomposition.

Partial Fraction Using Ti 89

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Though, partial fraction decomposition techniques seem to be easy and quick. They are so for many fractions. But for several other complicated rational fractions, just few techniques of partial fractions are insufficient. Sometimes, the given rational fractions are so complicated that they make the whole process difficult and a tedious job.

When partial fractions are required in order to solve other problems, they can be done by using scientific calculators. The calculators used for partial fractions are very advance and are able to perform such operations. Ti - 89 is a type of calculator that is most commonly used in order to find partial fraction decomposition of a rational fraction. One can easily determine partial fractions by using Ti - 89. But, it is just an alternative. The students must know and practice the process of partial fraction decomposition by their own.

Integration by Partial Fraction

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There are several integrals in which rational fractions are involved. These integrals usually cannot be integrated using other techniques. One needs to use partial fraction decomposition in order to integrate those. Generally, rational fractions are in such a way that there is no other technique of integration can be applied. These types of integrands should be first decomposed by using partial fractions and then integrated using suitable integration methods.

For Example:

$\int$ $\frac{3x+11}{x^{2}-x-6}$ $dx$

= $\int$  $\frac{3x+11}{(x-3)(x+2)}$ $dx$

The expression $\frac{3x+11}{(x-3)(x+2)}$ is to be decomposed using partial fractions.

$\frac{3x+11}{(x-3)(x+2)}$ = $\frac{A}{x-3}$ + $\frac{B}{x+2}$

$3x+11=A(x+2)+B(x-3)$

Substituting x = -2, we get

B = -1

Substituting x = 3, we get

A = 4

So,

$\frac{3x+11}{(x-3)(x+2)}$ = $\frac{4}{x-3}$ - $\frac{1}{x+2}$

Hence, we have

$\int$ $\frac{3x+11}{x^{2}-x-6}$ dx=$\int$  $\frac{4}{x-3}$ - $\frac{1}{x+2}$ $dx$

= 4 log (x - 3) - log (x + 2) + C

Examples

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Few examples of partial fraction decomposition are given below :

Example 1: Compute the partial fractions for the following:

$\frac{3x+5}{x^{2}+3x+2}$

Solution: $\frac{3x+5}{x^{2}+3x+2}$

= $\frac{3x+5}{(x+1)(x+2)}$

On representing this expression in the form of partial fractions:

$\frac{3x+5}{(x+1)(x+2)}$ =$\frac{A}{x+2}$+$\frac{B}{x+1}$

$3x+5=A(x+2)+B(x+1)$

On substituting x = -1, we get

- 3 + 5 = A (-1 + 2)

A = 2

Again, substituting x = -2, we get

-6 + 5 = B(-2+1)

B = 1

Therefore,

$\frac{3x+5}{(x+1)(x+2)}$ = $\frac{2}{x+1}$ + $\frac{1}{x+2}$


Example 2: Find the partial fractions for the expression $\frac{3x}{(x-1)^{2}}$.

Solution: $\frac{3x}{(x-1)^{2}}$

According to the rule, the form of partial fractions for the above expression would be:

$\frac{3x}{(x-1)^{2}}$ =$\frac{A}{x-1}$+$\frac{B}{(x-1)^{2}}$

3 x = A (x - 1) + B  _______(1)

Solving the above equation further:

3 x = A (x - 1) + B

3 x = A x - A + B

3 x + 0 = A x + (-A + B)

Equating the coefficient of x and constants on both the sides

A = 3

and

-A + B = 0

-3 + B = 0

B = 3

Therefore,

$\frac{3x}{(x-1)^{2}}$=$\frac{3}{x-1}$+$\frac{3}{(x-1)^{2}}$