charmedbeauty said:Homework Statement
Find ∫18/((x2+9)(x-3))
Homework Equations
The Attempt at a Solution
Im a little stuck on this.
18∫1/((x2+9)(x-3))
Im not sure how to turn this into a partial fraction.. help.
Thanks
Curious3141 said:The general form for the PF decomposition would be:
[tex]\frac { 18 }{ (x^{ 2 }+9)(x-3) } =\frac { Ax+B }{ x^{ 2 }+9 } +\frac { C }{ x-3 } [/tex]
First find C by using the Heaviside coverup rule (put x = 3 after multiplying both sides by (x-3).
Then just subtract the [itex]\frac { C }{ x-3 } [/itex] from the LHS and simplify to find the remaining term.
A partial fraction problem is a mathematical problem that involves breaking down a rational function into simpler fractions. This is done by finding the partial fractions that make up the original function, which can then be used to solve the problem.
To solve a partial fraction problem, you first need to factor the denominator of the rational function into linear and/or irreducible quadratic factors. Then, you set up and solve a system of equations to find the coefficients of the partial fractions. Once you have the partial fractions, you can combine them and solve for the original function.
Solving partial fraction problems can help simplify complex rational functions and make them easier to work with. It is also useful in integration, as it allows for the use of simpler integration techniques on the partial fractions rather than the original function.
Yes, there are a few special cases that may arise when solving partial fraction problems. These include repeated linear factors, quadratic factors with complex roots, and quadratic factors with repeated roots. Each of these cases requires a slightly different approach to finding the partial fractions.
Sure, let's say we have the rational function f(x) = (3x+1)/(x^2 + 5x + 6). To solve this partial fraction problem, we first factor the denominator into (x+2)(x+3). Then, we set up the system of equations: A/(x+2) + B/(x+3) = (3x+1)/(x+2)(x+3). Solving this system of equations gives us A = 1 and B = 2. Therefore, the partial fraction decomposition of f(x) is (1/(x+2)) + (2/(x+3)).