# Integrate x/(x^2+2)^2 ?

Apr 4, 2017

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

#### Explanation:

We use the substitution rule. We define a new variable $u$ as $u = {x}^{2} + 2$. This implies that $\mathrm{du} = 2 x \setminus \mathrm{dx}$, or $\mathrm{dx} = \frac{\mathrm{du}}{2 x}$.

We substitute these values in our original problem $\int \setminus \frac{x}{{x}^{2} + 2} ^ 2 \setminus \mathrm{dx}$ to get $\int \setminus \frac{x \setminus \mathrm{du}}{2 x {u}^{2}}$. Canceling out the $x$'s, we get $\int \setminus \frac{\mathrm{du}}{2 {u}^{2}}$.

This becomes a simple matter of integrating $\frac{1}{2} {u}^{-} 2$ with respect to $u$. Using the constant rule and power rule in integration, we obtain $- 1 \cdot \frac{1}{2} {u}^{-} 1 + C = - \frac{1}{2 u} + C$.

However, we want our answer in terms of $x$. Remember that we first defined $u$ as ${x}^{2} + 2$. We just need to substitute this back to get $- \frac{1}{2 \left({x}^{2} + 2\right)} + C$.