Integrating with Partial Fractions

What are partial fractions?

Partial fractions arise when a quotient is rewritten as the sum of fractions

The process is the opposite of adding or subtracting fractions

Each partial fraction has a denominator which is a linear factor of the quotient’s denominator

e.g. A quotient with a denominator of $x^{2} + 4 x + 3$

factorises to $(x + 1) (x + 3)$
so the quotient will split into two partial fractions
one with the (linear) denominator $(x + 1)$
one with the (linear) denominator $(x + 3)$

How do I know when to use partial fractions in integration?

For this course, the denominators of the quotient will be of quadratic form

i.e. $f (x) = a x^{2} + b x + c$
check to see if the quotient can be written in the form $\frac{f' (x)}{f (x)}$

in this case, reverse chain rule applies

If the denominator does not factorise then the inverse trigonometric functions are involved

How do I integrate using partial fractions?

STEP 1
Write the quotient in the integrand as the sum of partial fractions
This involves factorising the denominator, writing it as an identity of two partial fractions and using values of $x$ to find their numerators
e.g. $I = \int_{}^{} \frac{1}{x^{2} + 4 x + 3} d x = \int_{}^{} \frac{1}{(x + 1) (x + 3)} d x = \frac{1}{2} \int_{}^{} (\frac{1}{x + 1} - \frac{1}{x + 3}) d x$

STEP 2
Integrate each partial fraction leading to an expression involving the sum of natural logarithms
e.g. $I = \frac{1}{2} \int_{}^{} (\frac{1}{x + 1} - \frac{1}{x + 3}) d x = \frac{1}{2} [\ln |x + 1| - \ln |x + 3|] + c$

STEP 3
Use the laws of logarithms to simplify the expression and/or apply the limits
(Simplifying first may make applying the limits easier)
e.g. $I = \frac{1}{2} \ln |\frac{x + 1}{x + 3}| + c$

By rewriting the constant of integration as a logarithm ( $c = \ln k$ , say) it is possible to write the final answer as a single term
e.g. $I = \frac{1}{2} \ln |\frac{x + 1}{x + 3}| + \ln k = \ln \sqrt{|\frac{x + 1}{x + 3}|} + \ln k = \ln (k \sqrt{|\frac{x + 1}{x + 3}|})$

Examiner Tip

Always check to see if the numerator can be written as the derivative of the denominator
- If so then it is reverse chain rule, not partial fractions
- Use the number of marks a question is worth to help judge how much work should be involved

Worked example

Find $\int_{}^{} \frac{3 x + 1}{x^{2} + 3 x - 10} d x$ .

5-9-3-ib-hl-aa-only-we-soltn

Integrating with Partial Fractions (DP IB Maths: AA HL)

Revision Note