Resolve $\dfrac{x^2+1}{(x-1)^3}$ into partial fractions.

Resolve (x²+1)/((x-1)³) into partial fractions.

Exercise 7(a) · Q3

\dfrac{x^2+1}{(x-1)^3}

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Step 1. Let $t=x-1$ , so $x=t+1$ . Rewrite the numerator in terms of $t$ :

x^2+1=(t+1)^2+1=t^2+2t+1+1=t^2+2t+2.

Step 2. The fraction becomes

\frac{t^2+2t+2}{t^3}=\frac{t^2}{t^3}+\frac{2t}{t^3}+\frac{2}{t^3}=\frac1t+\frac2{t^2}+\frac2{t^3}.

Step 3. Substitute back $t=x-1$ :

$\frac{x^2+1}{(x-1)^3}=\frac1{x-1}+\frac2{(x-1)^2}+\frac2{(x-1)^3}.$ …