11TH•Mathematics•Partial Fractions•Question 13

Solution

Resolve into partial fractions: $\dfrac{1}{x(x+1)^3}$

Let:

\frac{1}{x(x+1)^3}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^2}+\frac{D}{(x+1)^3}

Multiply by $x(x+1)^3$ :

1=A(x+1)^3+Bx(x+1)^2+Cx(x+1)+Dx

Put $x=0$ :

1=A\Rightarrow A=1

Put $x=-1$ :

1=-D\Rightarrow D=-1

Solving remaining coefficients gives $B=-1$ , $C=-1$ .

Hence:

\boxed{\frac{1}{x(x+1)^3}=\frac{1}{x}-\frac{1}{x+1}-\frac{1}{(x+1)^2}-\frac{1}{(x+1)^3}}

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