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Evaluate the integral of (arctan x1/2)/(x1/2(1+x))

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*}

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