Integration by Partial Fractions

Thread starter RedBarchetta
Start date Sep 21, 2008
Tags

Fractions Integration Partial Partial fractions

In summary, integration by partial fractions is a method used to simplify and solve integrals involving rational functions. It is used when other integration techniques cannot be applied, and involves factoring the denominator, writing the function as a sum of simpler fractions, determining constants, and then integrating and simplifying. It cannot be used for all rational functions and has three special cases for repeated linear and irreducible quadratic factors in the denominator.

Sep 21, 2008

RedBarchetta

Homework Statement

[tex]\[
\int {\frac{{e^t dt}}
{{e^{2t} + 3e^t + 2}}}
\]
[/tex]

I'm not quite sure how to start this one...Any hints? I tried bringing e^t down to the denominator and multiplying it out which still didn't help. I can't see a way to factor the denominator or split this into a partial fraction.

Thank you.

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Sep 21, 2008

Defennder

Homework Helper

Why can't you factor the denominator? Denote x = e^t. Can you do it now?

Related to Integration by Partial Fractions

1. What is integration by partial fractions?

Integration by partial fractions is a method used to simplify and solve integrals that involve rational functions (functions with polynomials in the numerator and denominator). It involves breaking down a complex rational function into simpler fractions that can be easily integrated.

2. When is integration by partial fractions used?

Integration by partial fractions is used when integrating rational functions that cannot be easily solved using other integration techniques, such as substitution or integration by parts. It is also useful when solving integrals involving improper fractions (where the degree of the numerator is equal to or greater than the degree of the denominator).

3. What are the steps for integration by partial fractions?

The general steps for integration by partial fractions are:

Factor the denominator of the rational function into irreducible factors.
Write the rational function as a sum of simpler fractions, with one fraction for each irreducible factor in the denominator.
Determine the unknown constants in the numerator of each fraction by equating the coefficients of like terms on both sides of the equation.
Integrate each fraction separately.
Combine the integrals and simplify the result.

4. Can integration by partial fractions be used for all rational functions?

No, integration by partial fractions can only be used for proper rational functions (where the degree of the numerator is less than the degree of the denominator) and improper rational functions (where the degree of the numerator is equal to or greater than the degree of the denominator).

5. Are there any special cases in integration by partial fractions?

Yes, there are three special cases in integration by partial fractions:

Repeated linear factors in the denominator: In this case, the partial fractions will have the form A/ (x-a)^n, where n is the number of repeated factors and A is a constant.
Irreducible quadratic factors in the denominator: In this case, the partial fractions will have the form (Ax+B)/ (x^2 + bx + c), where A and B are constants determined by equating coefficients.
Repeated irreducible quadratic factors in the denominator: In this case, the partial fractions will have the form (Ax+B)/ [(x^2 + bx + c)^n], where n is the number of repeated irreducible factors and A and B are constants determined by equating coefficients.