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Mathematics 1A Tutorial
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Partial Fractions Tutorial EN1911 (Mathematics 1A)Tutorials– Partial Fractions
𝐴
Each simple linear factor (𝑥 + 𝑎) in the denominator contributes a partial fraction .
𝑥+𝑎
𝐴 𝐵
Each squared linear factor (𝑥 + 𝑎)2 in the denominator contributes partial fractions + (𝑥+𝑎)2 .
𝑥+𝑎
𝐴𝑥+𝐵
Each irreducible quadratic factor ((𝑥 + 𝑎)2 + 𝑏 2 ) in the denominator contributes a partial fraction .
((𝑥+𝑎)2 +𝑏 2 )
Express as Partial Fractions and hence integrate each function:
𝑥+7 10𝑥+37 8𝑠−28
1. 2. 3.
(𝑥−2)(𝑥−5) (𝑥−4)(𝑥+7) (𝑠−2)(𝑠−4)
For 4, 5 & 6, first factorise the denominator, then express as partial fractions and determine the
integral of each function:
3𝑥+5 𝑦−13 𝑥−14
4. 5. 6.
𝑥 2 +2𝑥−3 𝑦 2 −𝑦−6 𝑥 2 −10𝑥+24
Express as Partial Fractions and hence integrate each function:
17𝑥 2 −21𝑥−6 6𝑥 2 +7𝑥−49
7. 8.
𝑥(𝑥+1)(𝑥−3) (𝑥−4)(𝑥+1)(2𝑥−3)
In 9 & 10, first express as a polynomial plus proper algebraic fraction, then as partial fractions:
𝑥 2 +2 2𝑥 3 +7𝑥 2 −2𝑥−27
9. 10.
𝑥 2 +2𝑥−8 (𝑥−1)(𝑥+4)
For the following expressions with repeated linear factors, express as Partial Fractions:
2𝑡−1 1 9𝑥 2 −73𝑥+150
11. 12. 13.
(𝑡+1)2 𝑥 2 (𝑥+2) (𝑥−7)(𝑥−3)2
For the following expressions with irreducible quadratic factors, express as Partial Fractions:
1 3−𝑥 5𝑥 2 +7𝑥+17
14. 15. 16.
𝑥(𝑥 2 +1) (𝑥+3)(𝑥 2 +3) (𝑥+1)(𝑥 2 +2𝑥+6)
Express as Partial Fractions:
7 𝑥 3 +𝑥+1 5𝑥 2 +23𝑥+24
17. 18. 19.
(𝑥−2)(𝑥 2 +𝑥+1) 𝑥 2 +3𝑥+2 (2𝑥+3)(𝑥+2)2
Answers
4 3 𝐶(𝑥−5)4 7 3
1. − 𝑥−2, ln (𝑥−2)3
2. + 𝑥+7, ln 𝐶(𝑥 − 4)7 (𝑥 + 7)3
𝑥−5 𝑥−4
6 2 2 1
3. + 𝑠−4, ln 𝐶(𝑠 − 2)6 (𝑠 − 4)2 4. + 𝑥+3, ln 𝐶(𝑥 − 1)2 (𝑥 + 3)
𝑠−2 𝑥−1
3 2 𝐶(𝑦+2)3 5 4 𝐶(𝑥−4)5
5. − , ln (𝑦−3)2
6. − , ln (𝑥−6)4
𝑦+2 𝑦−3 𝑥−4 𝑥−6
2 8 7 3 2 4 𝐶(𝑥−4)3 (2𝑥−3)2
7. + 𝑥+1 + 𝑥−3 , ln 𝐶𝑥 2 (𝑥 + 1)8 (𝑥 − 3)7 8. − 𝑥+1 + , ln (𝑥+1)2
𝑥 𝑥−4 2𝑥−3
3 1 4 7 2 3 1 1 1
9. 1 − 𝑥+4 + 𝑥−2 10. 2𝑥 + 1 − 𝑥−1 + 𝑥+4 11. − (𝑡+1)2 12. − 4𝑥 + 4(𝑥+2)
𝑡+1 2𝑥 2
5 4 3 1 𝑥 1−𝑥 1 3 2𝑥−1
13. + 𝑥−3 − (𝑥−3)2 14. − 𝑥 2 +1 15. + 16. + (𝑥+1)2 +5
𝑥−7 𝑥 2(𝑥 2 +3) 2(𝑥+3) 𝑥+1
1 (𝑥+3) 1 9 3 1 2
17. (𝑥−2)
− (𝑥 2 +𝑥+1) 18. 𝑥 − 3 − 𝑥+1 + 𝑥+2 19. + 𝑥+2 + (𝑥+2)2
2𝑥+3
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