Answer
$\approx0.8527$
Work Step by Step
Here, $dr=(\cos t i-\sin t j+\sec^2 t k) dt$ and $F(r(t))=\sin te^{\sin t} i+\tan t \sin t e^{\cos t}j+\sin t\cos te^{\tan t} k$
$\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=\int_0^{\pi/4} (\sin te^{\sin t} i+\tan t \sin t e^{\cos t}j+\sin t\cos te^{\tan t} k) \cdot (\cos t i-\sin t j+\sec^2 t k) dt$
or, $=\int_0^{\pi/4} \sin t \cos t e^{\sin t} -\tan t \sin^2 t e^{\cos t}+\tan t e^{\tan t} dt$
By using calculator, we have $\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}\approx0.8527$