Integration Using Partial Fraction - Practice Questions & MCQ

A rational function is of the form P(x) / Q(x), where P(x) and Q(x) are polynomial function and Q(x) ≠ 0.

Partial fraction method is used to integrate rational functions.

Partial fraction decomposition can be applied to a rational function only if deg (P(x)) < deg (Q(x)).

In the case when deg (P(x)) ≥ deg(Q(x)), we must first perform long division to rewrite the quotient in the form of . where deg (R(x)) < deg (Q(x)). We then do a partial fraction decomposition on .

 

Illustration on how to approach to integrals of rational functions of the form where, deg (P(x)) ≥ deg(Q(x))

 

If the rational function  is proper (i.e. deg (P(x)) < deg (Q(x)) ), then three cases arises

 

Case 1: 

When denominator (Q(x)) is expressed as the product of the non-repeated linear factor

 

 

Short cut method

 

A, B, C  can now be determined by

Option 1 : comparing coefficients of different powers of x and the constant terms on both sides.

Option 2: Since the relation that we just obtained should held true for all x, we substitute those values of x that would straight way give us the required values of A, B and C. These values are obviously the roots of Q(x).

 

Case 2: 

When denominator (Q(x)) is expressed as the product of the linear factor such that some of them are repeating (Linear and repeated)

 

Case 3: 

When some of the factors in denominator (Q(x)) are quadratic but non-repeating 

Corresponding to each quadratic factor ax² + bx + c, assume the partial fraction of the type  

, where A and B are constants to be determined by comparing coefficients of similar power of x in the numerator of  both sides. or by substituting suitable values of x on both sides