Integration by partial fractions is a technique used to simplify the integration of rational functions. It allows us to break down a complex fraction into simpler fractions that can be integrated easily.
You can use integration by partial fractions when the degree of the numerator is less than the degree of the denominator of a rational function. It can also be used when the denominator can be factored into distinct linear or quadratic factors.
The steps for integrating by partial fractions include:
1. Write the rational function in the form of a sum of simpler fractions.
2. Determine the unknown coefficients by equating the numerators of the simpler fractions with the original numerator.
3. Write the resulting fractions as separate integrals.
4. Evaluate each integral and combine them to get the final answer.
Yes, integration by partial fractions can be used for improper fractions. The resulting simpler fractions may have polynomials with higher degrees, but the integration can still be performed using the same steps as for proper fractions.
There are a few restrictions for using integration by partial fractions. The rational function must be proper, meaning that the degree of the numerator must be less than the degree of the denominator. Additionally, the denominator must be factorable into distinct linear or quadratic factors. If these conditions are not met, other techniques may need to be used for integration.