What is the antiderivative of x/(1-x)^4?

1 Answer
George C.
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

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