Evaluate the integral of (arctan x^1/2)/(x^1/2(1+x))

by RoRi

November 30, 2015

Evaluate the following integral

$\int \frac{\arctan \sqrt{x}}{\sqrt{x}(1+x)} \, dx.$

We make a substitution,

$u = \arctan \sqrt{x} \quad \implies \quad du = \frac{dx}{2 \sqrt{x} (1+x)}.$

Then we have

$\begin{align*} \int \frac{\arctan \sqrt{x}}{\sqrt{x} (1+x)} \, dx &= 2 \int u \, du \\ &= u^2 + C \\ &= (\arctan \sqrt{x})^2 + C. \end{align*}$

Stumbling Robot