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CIE A-Level Maths Study Notes

2.5.3 Partial Fractions in Integration

Mastering the technique of partial fractions is a cornerstone when integrating complex rational functions. This method simplifies these functions into a sum of simpler fractions that are more straightforward to integrate.

Decomposition into Partial Fractions

To integrate a complex rational function:

1. Break it down into simpler fractions with linear or quadratic denominators using partial fractions.

2. Integrate each simpler fraction separately.

Example:

For 122x31x2(2x1)dx\int_{1}^{2} \frac{2x^3 - 1}{x^2(2x - 1)} dx:

Express as partial fractions:

2x31x2(2x1)=Ax+Bx2+C2x1\frac{2x^3 - 1}{x^2(2x - 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x - 1}

Integration After Decomposition

Integrate a function by partial fractions:

1. Decompose into simpler fractions.

2. Integrate each fraction.

Example:

For 122x31x2(2x1)dx\int_{1}^{2} \frac{2x^3 - 1}{x^2(2x - 1)} dx:

Decomposed form:

12(2x+1x2+32x1)dx\int_{1}^{2} \left( \frac{2}{x} + \frac{1}{x^2} + \frac{3}{2x - 1} \right) dx

Integrated result:

x+2lnx32ln2x1x + 2 \ln|x| - \frac{3}{2} \ln|2x - 1|

With limits applied:

log(439)+32\log\left(\frac{4\sqrt{3}}{9}\right) + \frac{3}{2}
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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