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 a partial fraction. Partial fractions have many uses (such as in integration). The method of partial fractions allows us to split the right-hand side of the above equation into the left-hand side. Let us understand more about partial fractions in the coming section.

Partial Fractions from Rational Functions

• Expression for the partial fraction formula:-Any number that can be represented as p/q easily, such that p and q are integers and where the value of q cannot be zero are known as Rational numbers. These are 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), where 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.

Proper Fraction of Improper Function

If the degree of the numerator is more than or equal to the denominator, the algebraic fraction is improper. The degree is the polynomial's highest power. Assume that m is the degree of denominator and n is the degree of numerator. Then, in addition to the partial fractions resulting from factors in the denominator, we must also include an additional term: a polynomial of degree n - m. Remember: A polynomial with zero degree is K, where K is a constant A polynomial of degree 1 is Px + Q A polynomial of degree 2 is P\[x^{2}\]+Qx+K

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?

First, 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.

• 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.

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!

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.

Tips To Split A Fraction into its Partial Fractions

To split a fraction into its partial fractions, use the following tips.

• If there are non-repeated linear factors in the denominator, the constants can be found by setting each linear factor to zero.

• If there are repeated linear factors and/or irreducible quadratic factors in the denominator:

• To get the value of some constants, set the linear factors to zero.

• To get at least one more constant, set x = 0.

• To find the other constants, compare the coefficients of \[X^{3}\],\[x^{2}\],..., etc.