When integrating functions involving polynomials in the denominator, partial fractions can be used to simplify integration. New students of Calculus will find it handy to learn how to decompose functions into partial fractions not just for integration, but for other aspects of Calculus as well, once they enter more advanced studies.
EditSteps
- Check to make sure that the fraction you are trying to integrate is proper. A proper fraction has a larger power in the denominator than in the numerator. If the power of the numerator is larger than or equal to the power of the denominator, it is improper and must be divided using long division.
- In this example, the fraction is indeed improper because the power of the numerator, 3, is larger than the power of the denominator, 2. Therefore, long division must be used.
- In this example, the fraction is indeed improper because the power of the numerator, 3, is larger than the power of the denominator, 2. Therefore, long division must be used.
- After dividing your improper fraction with long division, your new fraction can be put in the form of quotient + (remainder / divisor). In this example, we must use partial fractions to integrate the second fraction.
- Factor the polynomials in the denominator.
- Separate the fraction that you wish to decompose in to multiple fractions. The number of fractions in decomposition should equal the number of factors of x. The numerators of these decomposed fractions should be represented with constants.
- If a factor of x in the denominator has a power higher than 1, then the constants in the numerator should reflect this higher power. For example, x^2 in the denominator should be represented with the constant Ax + B.
- Multiply both sides by the denominator of the original fraction in order to get rid of all denominators. Notice that right now, the right side is factored by coefficients.
- On the side of the decomposed fractions, multiply everything out.
- Factor the decomposed side by the x terms.
- Set the coefficients equal on both sides. Because both sides are equal, that means that the coefficients of the x terms are equal.
- Solve for all constants.
- Plug the constants in to the decomposed fractions.
- If you divided earlier because the fraction you wished to integrate was improper, don't forget to plug in the rest of the integrand.
- Integrate.
EditTips
- Do not forget a + C if evaluating an indefinite integral!
EditRelated wikiHows
- How to Integrate by Substitution
- How to Integrate by Parts
- How to Integrate by Parts when One Function Is a Polynomial
- How to Differentiate Polynomials