Answer
$\left\langle \frac{\pi}{4}, \frac{1}{2}\ln(2) \right\rangle$
Work Step by Step
We have
\begin{align}
\int_{0}^{1}\left\langle \frac{1}{1+s^2},\frac{s}{1+s^2}\right\rangle ds&=\left\langle \tan^{-1}s, \frac{1}{2}\ln(s^2+1)\right\rangle|_{0}^{1}\\
&=\left\langle \tan^{-1}1, \frac{1}{2}\ln(2)\right\rangle-\left\langle \tan^{-1}0, \frac{1}{2}\ln(1)\right\rangle\\
&= \left\langle \frac{\pi}{4}, \frac{1}{2}\ln(2) \right\rangle.
\end{align}
