# What is the arclength of f(x)=arctan(2x)/x on x in [2,3]?

Jun 6, 2018

$1.01941$

#### Explanation:

Differentiaitng
$f \left(x\right) = \arctan \frac{2 x}{x}$ with respect to $x$
we get

$f ' \left(x\right) = \frac{2}{x \left(1 + 4 {x}^{2}\right)} - \arctan \frac{2 x}{x} ^ 2$
so we have to solve the integral
${\int}_{2}^{3} \sqrt{1 + {\left(\frac{2}{x \cdot \left(1 + 4 {x}^{2}\right)} - \arctan \frac{2 x}{x} ^ 2\right)}^{2}} \mathrm{dx} \approx 1.01941$