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Integration of Rational Functions by Partial Fractions

An expression of the type P(x) = a0xn + a1xn–1 + ... + an–1 x + an where a, a1, a2, ..., an are real numbers, a0 ≠ 0 and n is positive integer is called a polynomial of degree n.
 
A function of the form P/Q where P and Q are polynomials is called rational function. Consider the rational function 92361.png.
 
The two fractions on the RHS are called the partial fractions.
 
Any rational algebraic function is called a proper fraction, if the degree of numerator is less than of its denominator, otherwise it is called an improper fraction.
 
For example, 92355.png is a proper fraction, whereas
92349.png
is an improper fraction.
 
To integrate the rational function on the LHS, it is enough to integrate the two fractions on the RHS which is easy. This is known as the method of partial fractions. Here, we assume that the denominator can be fractional into linear or quadratic factors.

 

Note:
In using the method of partial fractions, we must have the degree of polynomial in numerator P(x) always less than that of denominator Q(x). If it is not so, we carry out the division of P(x) by Q(x) and reduce the degree of the numerator to less than that of the denominator, i.e.,
 
93182.png = P1(x) + 93176.png
 
where the degree of P2(x) < degree of Q(x).
 
Then to integrate, we apply the method of partial fractions to P2(x)/Q(x).

 

The partial fractions depend on the nature of the factors of Q(x). We have to deal with the following different types:
 
Form of the rational function
Form of the partial fraction
92317.png
92311.png
92304.png
92297.png
92290.png
92284.png
92278.png
92272.png
92266.png
92260.png
where x2 + bx + c cannot be factorized further
 
We find the values of constants A, B, and C by taking LCM and cross-multiplying and comparing the coefficients of different powers of x.




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