Integration with Partial Fractions

Basic Concepts

Understand the concept of partial fractions, which involves breaking simpler fractions up into a sum of fractions.
These fractions, known as partial fractions, are less complex to work with. Particularly handy when integrating or finding the original function.
Recognise different types of decompositions based on the given polynomial. These include proper fractions, improper fractions, and fractions involving repeated and non-repeated linear factors or quadratic factors.
The numerator’s degree should be lower than the denominator for proper fractions. If the numerator’s degree is equal or more, it’s an improper fraction.

Become familiar with utilizing the method of partial fractions in integration, to break complex rational functions into simpler fractions, thereby simplifying the process of integration.
Remember that the integrals of 1/x and x^n (n ≠ -1) are ln x and x^(n+1) / (n+1) respectively. These are very handy when dealing with partial fractions.
Integrate partial fractions by recognising the general form they reduce to, such as 1/x, x^n, etc.
After decomposing into partial fractions, integrate each fraction separately.

Grasp the process of decomposition into partial fractions.
Express the given fraction as the sum of its partial fractions, then equate this to the original fraction to obtain a system of equations.
Solving these equations provides the constants for the partial fractions.

Understand the importance of partial fractions in real-world applications, such as solving differential equations, evaluating complex integrals in physics and engineering, simplifying circuits in electronics, etc.
Use of partial fraction decomposition simplifies the calculation process and allows for easier integration.
The entire process enhances the understanding of the behaviour of rational functions. Topological understanding is thus also improved.