Answer
$$\dfrac{\pi}{4}$$
Work Step by Step
$\int_CF.dr=\int_0^{\pi}(1+t)sintcostdt+\int_0^{\pi}sin^2tdt$
$=[\frac{-1}{2}(1+t)cos2t+\frac{sin2t}{4}]_0^{\pi}+[\frac{t}{2}-\frac{sin2t}{4}]_0^{\pi}$
$=-\dfrac{\pi}{4}+\dfrac{\pi}{2}$
$$\int_CF.dr=\dfrac{\pi}{4}$$
