integrate (x^2+8)/(x^2-5x+6) by partial fractions method.
I first did long division, and came up with 1+((5x+2)/(x^2-5x+6)). Then I did partial fraction for (5x+2)/(x^2-5x+6) by going A/(x-3)+B/(x-2) and I got 17/(x-3)-12/(x-2) to take the integral of. But when I check online with wolfram, the answer is different, with what i got: x+ 17ln(x-3)-12ln(x-2)+c.
Please explain and help step by step. Thanks!
1 - 12/(x-2) + 17/(x-3)
You lost the 1.
Other than that, it's ok.
I actually didn't lose the 1, if you look at my final answer.
Right. I misread you r text.
But, wolfram's answer looks the same to me, if you allow that
∫ du/u = ln|u|
To integrate the given expression using partial fractions, you have correctly identified that the numerator of the fraction, (x^2 + 8), can be factored as (x-3)(x-2).
Let's start from the beginning and go through the steps to properly integrate the expression.
Step 1: Simplify the expression using long division
Divide (x^2 + 8) by (x^2 - 5x + 6), and you correctly obtained the quotient as 1 and the remainder as (5x + 2).
Now, your expression becomes:
Integral of (1 + (5x+2)/(x^2-5x+6)) dx
Step 2: Express the fraction using partial fractions
To express the fraction (5x + 2)/(x^2 - 5x + 6) in partial fractions, you correctly set it up as:
(5x + 2)/(x^2 - 5x + 6) = A/(x-3) + B/(x-2)
Now, we need to find the values of A and B.
Step 3: Find the values of A and B
To find the values of A and B, you can either compare the numerators or multiply through by the common denominator (x-3)(x-2).
Multiply both sides of the equation by (x-3)(x-2), and you get:
5x + 2 = A(x-2) + B(x-3)
Expand the equation:
5x + 2 = (A + B)x - 2A - 3B
Now, equate the coefficients of x and the constant terms:
A + B = 5 --> Equation 1
- 2A - 3B = 2 --> Equation 2
To solve these two equations, you can either use substitution or elimination method. In this case, it is convenient to use elimination:
Multiply Equation 1 by 3:
3A + 3B = 15 --> Equation 3
Add Equation 2 and Equation 3:
-2A - 3B + 3A + 3B = 2 + 15
A = 17
Now, substitute the value of A into Equation 1:
17 + B = 5
B = -12
Therefore, A = 17, and B = -12.
Step 4: Integrate the partial fractions
You correctly obtained the partial fraction decomposition as:
(5x + 2)/(x^2 - 5x + 6) = 17/(x-3) - 12/(x-2)
Now, let's integrate:
Integral of (17/(x-3) - 12/(x-2)) dx
= 17 ln|x-3| - 12 ln|x-2| + C
The constant "C" accounts for the integration constant.
So, your final answer is:
x + 17 ln|x-3| - 12 ln|x-2| + C
This matches the result you obtained from WolframAlpha with the additional term of 'x'.
I hope this helps clarify the steps and the correct integration result for you!