How do you use partial fraction decomposition to decompose the fraction to integrate #x^4/((x-1)^3)#?

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Answer 1 · 2015-08-30T16:13:40

First perform the division.

In order to use partial fraction decomposition we must have the degree of the numerator less than the degree of the denominator.

#x^4/((x-1)^3) = x^4/(x^3-3x^2+3x-1)#

# = x+ (3x^2-3x+1)/(x-1)^3#

To find the partial fraction decomposition of #(3x^2-3x+1)/(x-1)^3#, find #A, B " and", C# so that:

#A/(x-1)+B/(x-1)^2 + C/(x-1)^3 = (3x^2-3x+1)/(x-1)^3#

Clear the denominators to get:

#A(x^2-2x+1)+B(x-1)+C = 3x^2-3x+1#

So #A=3# and

#-2A+B = -3#, so #B=3#

finally, #A-B+C=1#, so #C=1#

#x^4/((x-1)^3) = x+3/(x-1)-3/(x-1)^2 + 1/(x-1)^3#