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

For example

*x*3 etc…

(1 – *x*2) 1 + *x*2

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

∫(*x*^{5} + 2)/( *x*^{2} – 1) d*x*

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

*x*^{5} + 2 = (*x ^{2}*- 1) (

which implies

*x*^{5} + 2 = *x ^{3}* +

*x*^{2} – 1 *x*^{2}– 1

the fraction *x* + 2

*x*^{2} – 1

on partial decomposition gives ;

*x+* 2 = *x* + 2 B

*x ^{2}* – 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

*x*^{5} + = *x*^{2} + *x* – 1 3 1

*x*^{2} – 1 2 (*x* + 1) 2 (*x* – 1)

Now integrating,

∫(*x*^{5} + 2) / (*x*^{2} – 1) d*x* = ∫(*x*^{2} + *x*) d*x * - ∫(1/2) (1/*x* + 1) d*x* + ∫(3/2) (1/*x* – 1) d*x*

Since we have

∫(*x*^{2} + *x*) d*x* = *x*^{4} + *x*^{2}

4 2

∫(1/*x* +1) d*x *= ln |*x* + 1|

And,

(1/*x* + 1) d*x* = ln|*x* – 1|

we get

∫ (*x*^{5} + 2)/(*x*^{2} – 1) d*x* = *x*^{4} + *x*^{2} – 1 ln|*x* + 1| + 3 ln|*x* – 1|

4 2 2 2

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