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

Question

Elijah Abram · Accepted Answer

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#

Samantha Adkison · Answer

undefined To decompose the fraction $ \frac{x^4}{(x-1)^3} $ using partial fraction decomposition, we first factor the denominator $ (x-1)^3 $ to its linear factors. In this case, it's $ (x-1)(x-1)(x-1) $. Since the degree of the numerator $ x^4 $ is greater than the degree of the denominator $ (x-1)^3 $, we need to perform polynomial division first.

After dividing $ x^4 $ by $ (x-1)^3 $, we get a quotient and a remainder. The quotient will be a polynomial of degree less than the denominator, which is $ (x-1)^3 $. The remainder will be a polynomial of degree less than the divisor $ (x-1)^3 $.

After performing polynomial division, the expression $ \frac{x^4}{(x-1)^3} $ can be rewritten as a sum of partial fractions, each with a simpler denominator than the original. The partial fraction decomposition will look like $ \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3} $.

We then find the values of constants $ A $, $ B $, and $ C $ by equating the original expression with the sum of partial fractions and solving for $ A $, $ B $, and $ C $ using algebraic methods, such as equating coefficients.

Once we have found the values of $ A $, $ B $, and $ C $, we substitute them back into the partial fraction decomposition.

Finally, we integrate each partial fraction term separately to obtain the integral of $ \frac{x^4}{(x-1)^3} $.