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 $\int_{1}^{2} \frac{2x^3 - 1}{x^2(2x - 1)} dx$:

Express as partial fractions:

$\frac{2x^3 - 1}{x^2(2x - 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x - 1}$## Integration After Decomposition

Integrate a function by partial fractions:

1. Decompose into simpler fractions.

2. Integrate each fraction.

#### Example:

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

Decomposed form:

$\int_{1}^{2} \left( \frac{2}{x} + \frac{1}{x^2} + \frac{3}{2x - 1} \right) dx$Integrated result:

$x + 2 \ln|x| - \frac{3}{2} \ln|2x - 1|$With limits applied:

$\log\left(\frac{4\sqrt{3}}{9}\right) + \frac{3}{2}$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.