How do you integrate #int (x-3)/(x^2-2x-5) dx# using partial fractions?
1 Answer
# int \ (x-3)/(x^2-2x-5) \ dx = 1/2ln |x^2-2x-5| -1/sqrt(6)ln | (x-1-sqrt(6))/(x-1+sqrt(6)) | + C #
Explanation:
We seek:
# I = int \ (x-3)/(x^2-2x-5) \ dx #
If we look at the quadratic on the numerator, we finds that if
# x^2-2x-5 = 0 => x = 1 +- sqrt(6) #
Therefore we find that we do not get "perfect" factors but rather:
# (x^2-2x-5 ) = (x - 1 - sqrt(6))(x - 1 + sqrt(6))#
So, although we could decompose into partial fractions it actually complicates the problem with the risk of an algebraic error.
A another approach, is to complete the square of the denominator:
# I = int \ (x-3)/((x-1)^2-1-5) \ dx #
# \ \ = int \ (x-3)/((x-1)^2-6) \ dx #
Substitute
# I = int \ (u-2)/(u^2-6) \ du #
# \ \ = int \ (u)/(u^2-6) - 2/(u^2-6) \ du #
# \ \ = int \ 1/2(2u)/(u^2-6) - 2/(u^2-sqrt(6)^2) \ du #
Both of these integrals are standard, so we can now integrate, giving:
# I = 1/2ln |u^2-6| -2 1/(2sqrt(6))ln | (u-sqrt(6))/(u+sqrt(6)) | + C #
# \ \ = 1/2ln |u^2-6| -1/sqrt(6)ln | (u-sqrt(6))/(u+sqrt(6)) | + C #
And, restoring the earlier substitution:
# I = 1/2ln |(x-1)^2-6| -1/sqrt(6)ln | ((x-1)-sqrt(6))/((x-1)+sqrt(6)) | + C #
# \ \ = 1/2ln |x^2-2x-5| -1/sqrt(6)ln | (x-1-sqrt(6))/(x-1+sqrt(6)) | + C #