# Integrate ((x / (1+x²)) ?

##### 1 Answer
Mar 17, 2018

$\frac{1}{2} \ln | 1 + {x}^{2} | + C$

#### Explanation:

We have: $\int$ $\frac{x}{1 + {x}^{2}}$ $\mathrm{dx}$

Let $u = 1 + {x}^{2} R i g h t a r r o w \mathrm{du} = 2 x$ $\mathrm{dx}$:

$= \frac{1}{2} \int$ $\frac{x}{1 + {x}^{2}} \cdot 2$ $\mathrm{dx}$

$= \frac{1}{2} \int$ $\frac{1}{1 + {x}^{2}} \cdot 2 x$ $\mathrm{dx}$

$= \frac{1}{2} \int$ $\frac{1}{u} \cdot$ $\mathrm{du}$

$= \frac{1}{2} \cdot \ln | u | + C$

We can now replace $u$ with $1 + {x}^{2}$:

$= \frac{1}{2} \ln | 1 + {x}^{2} | + C$