-3x^3+8x^2+24x-39 /(3x-2)^2 x (x^2-2x+5) integrate by partial fraction decomposition
What a cumbersome and ridiculous question!
First of all, due to your lack of brackets, I will assume you meant
(-3x^3+8x^2+24x-39) / ((3x-2)^2 * (x^2-2x+5) )
So you need this:
(-3x^3+8x^2+24x-39) / ((3x-2)^2 * (x^2-2x+5) )
= A/(3x-2) + (Bx+C)/(3x-2)^2 + (Dx+E)/(x^2 - 2x + 5)
Multiply each term by (3x-2)^2 (x^2 - 2x + 5)
then create 5 equations in 5 variables, using
x = 2/3, x=0, x=1, x=-1, x=2
I did this on paper but I am not going to type all that out.
Here is what Wolfram got and it agrees with mine:
www.wolframalpha.com/input?i=partial+fraction+%28-3x%5E3%2B8x%5E2%2B24x-39+%29%2F%28%283x-2%29%5E2+%28x%5E2-2x%2B5%29%29
Now enjoy integrating those 3 terms.
Here is what Wolfram said about that:
www.wolframalpha.com/input?i=%E2%88%AB+%28-3x%5E3%2B8x%5E2%2B24x-39+%29%2F%28%283x-2%29%5E2+%28x%5E2-2x%2B5%29%29+dx
I'm assuming you found C=0
You only need a linear numerator when the denominator is an irreducible quadratic.
actually, I meant B=0
:-(
To integrate the given expression using partial fraction decomposition, follow these steps:
Step 1: Factorize the denominator
The denominator of the expression is (3x-2)^2 × (x^2-2x+5). We need to factorize it completely before proceeding.
Step 2: Express the fraction as a sum of partial fractions
Write the given expression as a sum of partial fractions. Let's assume that the expression can be expressed as:
-3x^3+8x^2+24x-39 / [(3x-2)^2 × (x^2-2x+5)] = A/(3x-2) + B/(3x-2)^2 + (Cx + D)/(x^2-2x+5)
Step 3: Find the values of A, B, C, and D
To find the values of A, B, C, and D, we need to equate the numerators of the original expression to the numerators of the partial fractions.
-3x^3+8x^2+24x-39 = A(x^2-2x+5) + B + (Cx + D)(3x-2)
Simplify the equation and group like terms. Then compare the coefficients of the same powers of x on both sides to find the values of A, B, C, and D.
Step 4: Solve the system of equations
For (x^2-2x+5), equate the coefficients of x^2 and the constant term.
-3 = A
For (3x-2)^2, equate the coefficients of (3x-2)^2 and (3x-2).
8x^2 + 24x - 39 = B + (Cx +D)(3x-2)
Expand (Cx+D)(3x-2), and then compare coefficients of x^2, x, and the constant term.
Solve the system of equations to obtain the values of B, C, and D.
Step 5: Integrate each partial fraction
Once you have found the values of A, B, C, and D, you can integrate each partial fraction separately.
Integrate A/(3x-2) with respect to x using the formula for the integral of 1/u:
∫ (A/(3x-2)) dx = A * ln|3x-2| + C1, where C1 is the constant of integration.
Integrate B/(3x-2)^2 with respect to x using the formula for the integral of 1/u^2:
∫ (B/(3x-2)^2) dx = -B/(3x-2) + C2, where C2 is the constant of integration.
Integrate (Cx + D)/(x^2-2x+5) using partial fractions or another appropriate method.
Step 6: Combine the integrals
Finally, combine the integrals and include the appropriate constants of integration, C1 and C2, to obtain the final solution.
Remember to add '+ C3' at the very end, where C3 is the constant of integration that accounts for any remaining unknown constant terms in the original expression.
Note: The process of finding the values of A, B, C, and D might involve solving simultaneous equations, which can sometimes be complex.