Solutions - 0060-08

(1) Write the partial fractions general form equation:

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

Observe that $(x-1)$ appears in degree 3 in the integrand, so we have one term for each power up to 3 in the partial fraction decomposition.

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(2) Solve for constants:

Cross multiply:

$$1=A{(x-1)}^3+Bx{(x-1)}^2+Cx(x-1)+Dx$$

Plug in $x=0$, obtain $1=A(-1)$ so $A=-1$.

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

Plug in $x=2$, obtain:

$$1=(-1)\cdot 1+B\cdot 2+C\cdot 2+1\cdot 2\gg \gg 0=B+C$$

Plug in $x=-1$, obtain:

$$\begin{array}{c}1=(-1)(-8)+B(-1)\cdot 4+C(-1)(-2)+1\cdot (-1)\\ \\ \gg \gg -6=-4B+2C\\ \\ \gg \gg -6=+4C+2C\\ \\ \gg \gg C=-1,B=+1\end{array}$$

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(3) Integrate each term:

$$\begin{array}{c}\int \frac{1}{x{(x-1)}^3}dx\\ \\ \gg \gg \int \frac{-1}{x}dx+\int \frac{1}{x-1}dx+\int \frac{-1}{{(x-1)}^2}dx+\int \frac{1}{{(x-1)}^3}dx\\ \\ \gg \gg -\ln |x|+\ln |x-1|+\frac{1}{x-1}-\frac{1}{2{(x-1)}^2}+C\end{array}$$

Optional simplification:

$$\gg \gg \ln \left|\frac{x-1}{x}\right|+\frac{2x-3}{2{(x-1)}^2}+C$$