SOLUTION: Using the method of partial fractions, determine an antiderivative for this function: (x^2 + 1) / (x^3 - x^2 - 6x) The answer is supposed to be: -1/6 ln x + 2/3 ln | x - 3 | +

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Using the method of partial fractions, determine an antiderivative for this function: (x^2 + 1) / (x^3 - x^2 - 6x) The answer is supposed to be: -1/6 ln x + 2/3 ln | x - 3 | +       Log On


   

Question 564861: Using the method of partial fractions, determine an antiderivative for this function:
(x^2 + 1) / (x^3 - x^2 - 6x)
The answer is supposed to be: -1/6 ln x + 2/3 ln | x - 3 | + 1/2 ln | x + 2 | = C
How to do this? Thank you
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
First decompose the given fraction %28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29 into its partial fractions:
Factor the denominator:
%28x%5E2%2B1%29%2F%28x%29%28x%2B2%29%28x-3%29 now we set:
%28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29+=+A%2Fx%2BB%2F%28x%2B2%29%2BC%2F%28x-3%29 Now add the fractions on the right side.
Now since the denominators are equal, the numerators must be equal, so we can set:
x%5E2%2B1+=+A%28x%2B2%29%28x-3%29%2BB%28x%29%28x-3%29%2BC%28x%29%28x%2B2%29 This must be true for all x so it is true for the x-values that make:
x+=+0
x%2B2+=+0 so x+=+-2 and
x-3+=+0 so x+=+3
So we can solve for A, B, and C by letting x = 0, then x = -2, and x = 3.
After some standard algebra, you'll find that:
A+=+-1%2F6, B+=+-1%2F2, and C+=+2%2F3 we can now write the partial fraction:
%28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29+=+-1%2F6x-1%2F2%28x%2B2%29%2B2%2F3%28x-3%29
Now when you integrate these partial fractions you'll get the answers:
%28-1%2F6%29lnx-%281%2F2%29ln%28abs%28x%2B2%29%29%2B%282%2F3%29ln%28abs%28x-3%29%29%2BC