Answer
The partial fraction is, $\frac{1}{\left( x+1 \right)}-\frac{1}{{{\left( x+1 \right)}^{2}}}$
Work Step by Step
$\frac{x}{{{\left( x+1 \right)}^{2}}}=\frac{A}{\left( x+1 \right)}+\frac{B}{{{\left( x+1 \right)}^{2}}}$
Now, multiply both sides by ${{\left( x+1 \right)}^{2}}$:
${{\left( x+1 \right)}^{2}}\times \frac{x}{{{\left( x+1 \right)}^{2}}}={{\left( x+1 \right)}^{2}}\times \frac{A}{\left( x+1 \right)}+{{\left( x+1 \right)}^{2}}\times \frac{B}{{{\left( x+1 \right)}^{2}}}$
$\begin{align}
& x=A\left( x+1 \right)+B \\
& =Ax+A+B
\end{align}$
Then, compare the coefficient of $x$ and constant term:
$A=1$ …… (1)
$A+B=0$ …… (2)
Then, put the value of A in equation (2):
$\begin{align}
& 1+B=0 \\
& B=-1
\end{align}$
Therefore,
$\frac{x}{{{\left( x+1 \right)}^{2}}}=\frac{1}{\left( x+1 \right)}-\frac{1}{{{\left( x+1 \right)}^{2}}}$
Thus, the partial fraction of the given expression is $\frac{1}{\left( x+1 \right)}-\frac{1}{{{\left( x+1 \right)}^{2}}}$.
