# How do you find the integral of f(x) = arctan(x) ?

Sep 6, 2014

Integration by parts is a good place to start.

Let $y = u v$. Then $\int u \mathrm{dv} = u v - \int v \mathrm{du}$ - this is how to integrate by parts.

We need to choose a function u that is easily differentiated, and a function v that is easily antidifferentiated. For this, let's choose $v = x$ and $u = \arctan \left(x\right)$. Hence:
$\frac{\mathrm{dv}}{\mathrm{dx}} = 1 \implies \mathrm{dv} = \mathrm{dx}$
$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{1 + {x}^{2}} \implies \mathrm{du} = \frac{\mathrm{dx}}{1 + {x}^{2}}$

Now, we simply substitute these values in to our formula:
$\int u \mathrm{dv} = u v - \int v \mathrm{du}$
$\therefore$
$\int \arctan \left(x\right) \mathrm{dx} = \arctan \left(x\right) \cdot x - \int x \cdot \frac{\mathrm{dx}}{1 + {x}^{2}}$

From here, we can antidifferentiate the far right-hand term quite easily, and come up with an answer.