What is the antiderivative of #x/(1-x)^4#?
1 Answer
Jun 11, 2016
#int x/(1-x)^4 dx = (3x-1)/(6(1-x)^3) + C#
Explanation:
#x/(1-x)^4 = 1/(1-x)^3(x/(1-x))#
#= 1/(1-x)^3((1-(1-x))/(1-x))#
#= 1/(1-x)^3(1/(1-x)-1)#
#=1/(1-x)^4-1/(1-x)^3#
So:
#int x/(1-x)^4 dx = int 1/(1-x)^4-1/(1-x)^3 dx#
#= int (1-x)^(-4) - (1-x)^(-3) dx#
#= 1/3(1-x)^-3-1/2(1-x)^-2 + C#
#= 1/(3(1-x)^3)-1/(2(1-x)^2) + C#
#= 2/(6(1-x)^3)-(3(1-x))/(6(1-x)^3) + C#
#= (3x-1)/(6(1-x)^3) + C#
