Exam Questions - Integrals involving partial fractions

Integrals Involving Partial Fractions

When dealing with rational functions, where the numerator and the denominator are polynomials, partial fractions is a common technique used for breaking down complex fractions into simpler ones.
This technique is especially useful when dealing with integrations as it simplifies complex integrals into simpler, more manageable ones.

When a rational function is expressed as simpler fractions, it can be integrated term by term.
In cases with a linear factor in the denominator, the partial fraction is of the form A/x where A is a constant.
For repeated linear factors, the form is A/x + B/x² + C/x³ … and so forth.
With quadratic factors, it’s of form (Ax+B)/x² where A, B are constants and x is not divisible by the factor.

You start re-writing the integrand in terms of partial fractions, using algebraic division if necessary.
After expressing the integrand as simpler fractions, perform the integration term-by-term.
Remember to adjust the limits of the integral where necessary.

Firstly, determine the type of fraction.
Is it a proper fraction? (degree of the numerator is less than that of the denominator). If yes, you can proceed with breaking it into simpler fractions.
If not, you will need to perform algebraic division to rewrite it as a proper fraction.
After identifying the partial fraction structure, equate these fractions with the original fraction and compare coefficients to determine the constants for each term.
Lastly, perform the integration and apply the constant of integration if it’s an indefinite integral.

Start with ensuring the fraction is a proper fraction. If not, make it one using algebraic division.
Understand different partial fraction structures based on the factors present in the denominator.
Then, split the fraction into simpler fractions, calculate the constants, and perform the integration.
The integration of partial fractions simplifies many complex problems and understanding it thoroughly will always be in your advantage. Break it down into manageable chunks and don’t let those big fractions intimidate you!