Solutions - 0060-03

(1) Write the partial fractions general form equation:

$$\frac{1}{x{(x-1)}^2}\gg \gg \frac{A}{x}+\frac{B}{x-1}+\frac{C}{{(x-1)}^2}$$

---

(2) Solve for constants.

Cross multiply:

$$1=A{(x-1)}^2+Bx(x-1)+Cx$$

Plug in $x=1$, obtain $1=C$.

Plug in $x=0$, obtain $1=A$.

Plug in $x=2$, obtain:

$$\begin{array}{c}1=A+B\cdot 2+C\cdot 2\\ \\ \gg \gg 1=1+2B+2\gg \gg B=-1.\end{array}$$

---

(3) Integrate each term:

$$\begin{array}{cc}\int \frac{1}{x{(x-1)}^2}dx& \gg \gg \int \frac{1}{x}+\frac{-1}{x-1}+\frac{1}{{(x-1)}^2}\\ & \\ & \gg \gg \ln |x|-\ln |x-1|-{(x-1)}^{-1}+C\end{array}$$

Optional simplification:

$$\gg \gg \ln \left|\frac{x}{x-1}\right|-\frac{1}{x-1}+C$$