KTiaam said:Homework Statement
∫ 4x/(x^3+x^2+x+1) dx
The Attempt at a Solution
I really don't know where to start, you can't complete the square, the degree of the numerator is less than the denominator so you can't use long division to simplify it.
I can't really simplify the denominator as well, so I am stuck.
Help is greatly appreciated.!
Edit: I don't need you to work out the whole problem, i just need help getting started.
Integration by partial fractions is a method used to integrate rational functions, which are fractions with polynomials in the numerator and denominator. It involves breaking down a complex fraction into simpler fractions, making it easier to integrate.
This method is used when trying to integrate a rational function, especially when the degree of the numerator is less than the degree of the denominator. It is also used when the denominator of the fraction can be factored into linear and irreducible quadratic factors.
The first step is to factor the denominator of the rational function. Then, using the factored form, write the rational function as a sum of simpler fractions. Next, determine the unknown coefficients by equating the numerators of the simpler fractions to the original numerator. Finally, integrate each of the simpler fractions and combine the results to get the final solution.
Yes, there are a few restrictions to keep in mind. The denominator of the original rational function cannot have repeated linear factors, and all the irreducible quadratic factors must be distinct. Also, if the degree of the numerator is equal to or greater than the degree of the denominator, then the fraction cannot be broken down using this method.
Integration by partial fractions can be used in various engineering and physics applications, such as circuit analysis, control systems, and fluid mechanics. It can also be used in economics and finance to model and analyze interest rates and investment growth.