Why would you try long division? You are dividing something "smaller" into something "bigger". It is as if you are suggesting that to evaluate [itex]\frac{7}{584}[/itex] you divide 584 by 7.Cacophony said:I've foiled this out to look like:
∫(2x+1)/(x^4+3x³+4x²+3x+1)
I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or should I use a different approach?
Or when the degrees are equal.iRaid said:Definitely do partial fractions. The only time you do long division is when the degree on top is bigger than the bottom.
Long division in the context of integrals is a technique used to simplify complex integrals by breaking them down into smaller, more manageable parts. It involves using polynomial long division to divide the integrand by a simpler polynomial, which can then be integrated more easily.
Long division is typically used when the integrand is a rational function, meaning it is a ratio of two polynomials. This method is especially useful when the degree of the numerator is equal to or greater than the degree of the denominator. It can also be used when the integrand contains a quadratic expression.
To perform long division when integrating, follow these steps:
Using long division when integrating can greatly simplify the process and make it more manageable. It can also help identify patterns and relationships between terms, making it easier to find the antiderivative. Additionally, it can be used to find the partial fraction decomposition of a rational function, which is useful for solving more complex integrals.
Yes, there are some limitations to using long division in integration. This method is only applicable to rational functions, so it cannot be used for other types of integrals. It also may not work for all rational functions, as some may require other techniques such as substitution or integration by parts. Additionally, it can be time-consuming and may not always result in a simpler integral.