Answer
The partial fraction function is $-\frac{1}{x}+\frac{1}{\left( x-1 \right)}$.
Work Step by Step
We find the partial fraction of the provided expression as given below,
$\frac{1}{x\left( x-1 \right)}=\frac{A}{x}+\frac{B}{\left( x-1 \right)}$
Multiply $x\left( x-1 \right)$ on both sides,
$\begin{align}
& x\left( x-1 \right)\frac{1}{x\left( x-1 \right)}=x\left( x-1 \right)\frac{A}{x}+x\left( x-1 \right)\frac{B}{\left( x-1 \right)} \\
& 1=\left( x-1 \right)A+xB \\
& =Ax-A+Bx \\
& 1=x\left( A+B \right)-A
\end{align}$ …… (I)
Now, compare the coefficient of equation (I),
$A+B=0$ …… (II)
$\begin{align}
& -A=1 \\
& A=-1 \\
\end{align}$ …… (III)
Now, substitute the value of equation (III) into equation (II),
$\begin{align}
& -1+B=0 \\
& B=1
\end{align}$
Therefore,
$\frac{1}{x\left( x-1 \right)}=\frac{-1}{x}+\frac{1}{\left( x-1 \right)}$
Thus, the partial fraction of the provided expression is $\frac{-1}{x}+\frac{1}{\left( x-1 \right)}$.
