4.7 Integration Of Rational Functions:

A rational function is by definition the quotient of two polynomials.

For example

x3 etc…

(1 – x2) 1 + x2

are all rational functions. Remember in the rational function √x, ln(x) or |x| are not mentioned.

Steps to integrate the rational function

f(x) = P(x)

Q(x)

Step1: If degree(P) ≥ degree(Q) perform polynomial long-division. Otherwise go to step 2.

Step2: Factor the denominator Q(x) into irreducible polynomials: linear and irreducible quadratic polynomials.

Step3: Find the partial fraction decomposition.

Step4: Integrate the result of step 3.

Example: Find

∫(x⁵ + 2)/( x² – 1) dx

Solution. Since the degree of the numerator is higher than the denominator, we should perform the long-division. We get

x⁵ + 2 = (x²- 1) (x² + x) + x + 2

which implies

x⁵ + 2 = x³ + x + x +2

x² – 1 x²– 1

the fraction x + 2

x² – 1

on partial decomposition gives ;

x+ 2 = x + 2 B

x² – 1 (x – 1) (x + 1) x + 1 x - 1

Therefore,

x + 2 = A(x-1) + B(x+1).

If we substitute x=1, we get B = 3/2 and we substitute x= -1, we get A = -1/2. Therefore, we have

x⁵ + = x² + x – 1 3 1

x² – 1 2 (x + 1) 2 (x – 1)

Now integrating,

∫(x⁵ + 2) / (x² – 1) dx = ∫(x² + x) dx  - ∫(1/2) (1/x + 1) dx  + ∫(3/2) (1/x – 1) dx

Since we have

∫(x² + x) dx = x⁴ + x²

4 2

∫(1/x +1) dx = ln |x + 1|

And,

(1/x + 1) dx = ln|x – 1|

we get

∫ (x⁵ + 2)/(x² – 1) dx = x⁴ + x² – 1 ln|x + 1| + 3 ln|x – 1|

4 2 2 2

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