Integration using Partial Fractions (AQA A-Level Mathematics): Revision Notes

1. Factor the denominator into linear or quadratic terms.
2. Express the function as a sum of partial fractions.
3. Solve for constants in the partial fractions.
4. Integrate each term separately using basic integration rules. This method simplifies complex fractions, making them easier to integrate.

Example:

In this question, the idea is to notice the denominator can be factorised as a quadratic, then partial fractions can be used.

Now split into partial fractions:

Substitute values to find A and B:

5. Let $x = 4$ :

6. Let $x = -4$ :

Therefore:

Example: Write $\dfrac{20}{(x+3)(x^2+1)}$ in partial fractions.

7. Multiply through by $(x+3)(x^2+1)$:

8. Find constants A, B, and C:

• Let $x = -3$:

• Let $x = 0$:

• Let $x = 1$:

Example: By expressing $\dfrac{x^4 + x }{x^4 + 5x^2 + 6}$ in partial fractions, find

9. Multiply both sides by $(x^2+3)(x^2+2)$:

10. Expand both sides:

11. Combine like terms:

12. Set up equations by equating coefficients:

• $A + C = 0$
• $B + D = 5$
• $2A + 3C = -1$
• $2B + 3D = 6$

13. Solve the equations:

• From $A + C = 0, C = -A$.
• From $2A + 3C = -1$, substituting $C = -A$:

• From $B + D = 5$:

14. Integrate:

• Let $I = \int \dfrac{x+9}{x^2+3} dx + \int \dfrac{-x+4}{x^2+2} dx$
• $A = \frac{1}{2}\ln|x^2+3| + c_1$
• $B = \frac{9}{\sqrt{3}} \arctan\left(\frac{x}{\sqrt{3}}\right) + c_3$
• $C = \frac{1}{2} \ln|x^2+2| + c_2$
• $D = \frac{4}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right) + c_4$