Partial Fractions

It is possible to split many fractions into the sum or difference of two or more fractions, such a fraction is known as partial fraction. Partial fractions have many uses (such as in integration). The method of partial fractions basically allows us to split the right hand side of the above equation into the left hand side.


What are Partial Fractions and the Types of Partial Fractions?

This can be done directly by, using partial fraction formula :-

\[\frac{2}{x - 2} + \frac{3}{x + 1} \rightarrow \frac{5x - 4}{x^{2} - x - 2}\]

Here’s how you can do it, partial fractions examples and solutions :-

2x − 2 + 3x + 1 = 2(x + 1) + 3(x − 2) (x − 2)(x + 1)

We can simplify it using rational expressions:

= 2x + 2 + 3x − 6x2+ x − 2x − 2

= \[\frac{(5x − 4)}{(x^{2} − x − 2)}\]


Partial Fractions from Rational Functions

  • Expression for the partial fraction formula :-Any number that can be represented in the form of p/q easily , such that p and q are integers and where the value of q cannot be zero are known as Rational numbers. This is the partial fractions from rational functions.

  •  In the same manner, partial fractions from rational functions can be defined as the ratio of two polynomial functions P(x) and Q(x), here P and Q are polynomials in x and here Q(x) is not equal to 0. 

  • In Mathematics a rational function is known as proper if the degree of P(x) (top part of the fractions) is less than the degree of (low part of the fraction) that is Q(x) otherwise it is called an improper rational function. 

  • With the help of the long division polynomial process, we can reduce improper rational functions to proper rational functions.

  • Suppose , if the rational fraction is \[\frac{P(x)}{Q(x)}\] is improper then it can be expressed as this partial fraction formula:

\[\frac{P(x)}{Q(x)}\] can be written as A(x) +\[\frac{R(x)}{Q(x)}\]

A(x) in the above expression is a polynomial in x, where \[\frac{R(x)}{Q(x)}\] is said to be a proper rational function.

We know that the integration of a function f(x) is given by F(x) and we can represent it by:

\[\int f(x) dx = F(x)\] + C

Here the above of the equation represents the integral of f(x) with respect to x.


Here’s How to Solve Partial Fractions!

  1. Start with Proper Rational Expressions (if not, you need to division first).

  2.  You need to factor the bottom into linear factors. 

  3. Now you need to write out a partial fraction for each factor (and every exponent of each)

  4. Next, multiply the whole equation by the bottom.

  5. For solving the coefficients you need to substitute zeros of the bottoms.


Now What are Proper Rational Expressions?

Firstly, the concept of Partial Fractions works only in the case of proper rational expressions, A proper rational expression is one where the degree of the top is less than the bottom.


What is the Degree?

The degree can be defined as the largest exponent the variable has.

  • Proper fraction is one where the degree of the top is less than the degree of the bottom.


Proper Fraction:

\[\frac{x}{x^{3} - 1}\]

The degree of top of the given fraction is 1.
The degree of bottom of the given fraction is 3.

  • A partial fraction of Improper fraction is where the degree of the top is greater than, or equal to, the degree of the bottom in any given fraction.


Improper fraction:

\[\frac{x^{2} - 1}{x + 1}\]

The degree of top of the given fraction is 2.
The degree of bottom of the given fraction is 1


NOTE: If any given expression is in the form of a partial fraction or improper fraction, then you need to use polynomial division method first.


Partial Fractions Decomposition:

Partial fractions decomposition is an important concept in this topic.When we have to integrate a rational function, we need to reduce proper rational function. Partial fractions decomposition into a partial fraction method is a method in which the integrand is expressed as the sum of simpler rational functions. Now, after splitting the integrand into partial fractions, it is integrated according to integrating techniques.


The List of Types of Partial Fractions Formulas for Partial Fraction Methods is Given Below!


S.no

Rational Fraction

Partial Fraction

1.

\[\frac{p(x)+q}{(x - a)(x - b)}\]

\[\frac{A}{(x - a)} + \frac{B}{(x - b)}\]

2.

\[\frac{p(x)+q}{(x - a)^{2}}\]

\[\frac{A_{1}}{(x - a)}\] + \[\frac{A_{2}}{(x - a)^{2}}\]

3.

\[\frac{px^{2} + qx + r}{(x - a)(x - b)(x - c)}\]

\[\frac{A}{(x - a)} + \frac{B}{(x - b)} + \frac{C}{(x - c)}\]

4.

\[\frac{px^{2} + q(x) + r}{(x - a)^{2}(x - b)}\]

\[\frac{A_{1}}{(x - a)} + \frac{A_{2}} {(x - a)^{2}} + \frac{B}{(x - b)}\]

5.

\[\frac{px^{2} + qx + r}{(x - a)(x^{2} + bx + c)}\]

\[\frac{\frac{A}{(x - a)} + Bx + C}{(x^{2} + bx + c)}\]


In the above formulas, A, B, C are real numbers.


Partial Fractions Examples and Solutions (Integration):

Question 1) Solve the question given below using the concept of partial fractions.

                    (partial fraction integration example)

\[\int \frac{x}{(x + 2)(3 - 2x)} dx\]

Solution) Let's solve the given question using types of partial fractions,

From the partial fractions formula,

We can say I = \[\int \frac{x}{(x + 2)(3 - 2x)} dx\]

So now we get,

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

We get the following,

3 - 2x=0 

We now try to find the value of x ,

3 = 2x

Therefore, x= \[\frac{3}{2}\]

And A (0) +B (\[\frac{3}{2}\] + 2) = \[\frac{3}{2}\]

After further calculation we get,

B (\[\frac{7}{2}\]) = \[\frac{3}{2}\] where the value of B = \[\frac{3}{2}\]

When we further calculate,

B (\[\frac{7}{2}\]) = \[\frac{3}{2}\] where the value of B = \[\frac{3}{7}\]

Now, when we take x + 2 = 0 where x = -2

A (7)+B(0) = -2

Now, A = -\[\frac{2}{7}\]

We get,  \[\int \frac{x}{(x + 2)(3 - 2x)} dx\] = \[\frac{-2}{7}\] × \[\frac{1}{x + 2}\] + \[\frac{3}{7}\] × \[\frac{1}{3 - 2x}\]

This can be written as, \[\int \frac{x}{(x + 2)(3 - 2x)} dx\] = \[\int \frac{1}{x + 2} dx\] + \[\frac{3}{7} \int \frac{1}{3 - 2x} dx\] 

Now when we integrate with respect to x,

=\[\frac{-2}{7}\] log |x + 2|+ \[\frac{3}{7}\] × \[\frac{1}{-2}\]log|3-2x| + C

So now we get,

=\[\frac{-2}{7}\] log |x + 2|+\[\frac{3}{-14}\] log|3 - 2x| + C 

Above were the partial fractions examples and solutions for integration.

FAQ (Frequently Asked Questions)

Question 1)How do you Solve Partial Fractions and What is Partial Fraction Math?

Answer) Here’s how to solve partial fractions!

  • Start with Proper Rational Expressions (if not, you need to division first).

  • You need to factor the bottom into linear factors. 

  • Now you need to write out a partial fraction for each factor (and every exponent of each).

  • Next, multiply the whole equation by the bottom.

  • For solving the coefficients you need to substitute zeros of the bottoms.

  • Finally, now, you can write out your answer!

It is possible to split many fractions into the sum or difference of two or more fractions, such a fraction is known as partial fraction. Partial fractions have many uses (such as in the partial fraction integration example). The method of partial fractions basically allows us to split the right hand side of the above equation into the left hand side.

Question 2) What is the use of the Partial Fraction Method and What are Partial Fractions Used for and When can you use Partial Fraction?

Answer) The importance of the partial fraction decomposition generally  lies in the fact that partial fractions provide algorithms for various computations with rational functions, including the explicit computation of antiderivatives in Mathematics, Taylor series expansions, inverse Z-transforms and inverse Laplace transforms.


Partial Fractions can be used in Mathematics to turn functions that cannot be integrated into simple fractions easily. We can basically use partial fractions if the degree of the numerator is strictly less than the degree of the denominator.