- #1
whatlifeforme
- 219
- 0
Homework Statement
Integrate the following:
Homework Equations
∫(x^2+x-7)/(x+3)
The Attempt at a Solution
The only way I can think of solving this would be to split up each term into a separate fraction.
whatlifeforme said:Homework Statement
Integrate the following:
Homework Equations
∫(x^2+x-7)/(x+3)
The Attempt at a Solution
The only way I can think of solving this would be to split up each term into a separate fraction.
whatlifeforme said:how would i do that? i tried to separate the numerator into a product of two terms (x+3)(another term)... but that doesn't work.
whatlifeforme said:how would i do that? i tried to separate the numerator into a product of two terms (x+3)(another term)... but that doesn't work.
Ray Vickson said:Why not substitute u = x+3 and do a u-integration instead?
whatlifeforme said:so i would have:
1. ∫(x+3)(x-2) - 1 / (x+3)
2. ∫(x-2) - ∫1/(x+3)
whatlifeforme said:[i'm not sure how to take this integral though]
whatlifeforme said:so i would have:
1. ∫(x+3)(x-2) - 1 / (x+3)
2. ∫(x-2) - ∫1/(x+3) [i'm not sure how to take this integral though]
whatlifeforme said:i have for the answer:
x^2 - 2x - ln(x+3) + c
For this integration, Ray's suggestion appears to be most straight forward. (I realize that in general it's important to know how to do long division when integrating a rational expression for which the degree of the numerator is equal to or greater than hte degree of the denominator.)Ray Vickson said:Why not substitute u = x+3 and do a u-integration instead?
A rational function is a mathematical function that can be expressed as the quotient of two polynomial functions. It can be written in the form f(x) = p(x)/q(x), where both p(x) and q(x) are polynomial functions and q(x) is not equal to 0.
The process of integrating a rational function involves finding the indefinite integral of the function, which is the most general antiderivative of the function. This is done by using integration techniques such as substitution, integration by parts, or partial fractions.
A rational function is integrable if its denominator can be factored into linear and irreducible quadratic factors. If the denominator cannot be factored in this way, then the rational function is not integrable.
Integrating rational functions is important in many areas of mathematics and science. It helps in solving various problems involving rates of change, finding the area under curves, and calculating volumes of three-dimensional shapes. It is also used in fields such as physics, engineering, and economics.
Yes, there are some special cases when integrating rational functions. One such case is when the rational function has a constant in the numerator or denominator, in which case the integration can be simplified. Another special case is when the rational function has a repeated factor in the denominator, which requires a different integration technique called partial fractions.