Answer
$\int_0^1\langle\frac{1}{t+1},\frac{1}{t^2+1},\frac{t}{t^2+1}\rangle dt=\langle\ln{2},\frac{\pi}{4},\frac{\ln2}{2}\rangle$.
Work Step by Step
$\int_0^1\langle\frac{1}{t+1},\frac{1}{t^2+1},\frac{t}{t^2+1}\rangle dt=\int_0^1\frac{1}{t+1}dt+\int_0^1\frac{1}{t^2+1}dt+\int_0^1\frac{t}{t^2+1}dt=(\ln{(t+1)})\big|_0^1+(\arctan{t})\big|_0^1+(\frac{\ln{{(t^2+1)}}}{2})\big|_0^1=\langle\ln{2},\frac{\pi}{4},\frac{\ln2}{2}\rangle$
