Answer
$0.927136$
Work Step by Step
Here, we have $F[r(t)]=\sin te^{\sin t} i+\cos t \sin^2 t e^{\cos t}j+\sin t\cos te^{\tan t} k$ and $dr=(\cos t -\sin t j+\sec^2 t k) dt$
$\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=\int_0^{\pi/4} (\sin t(e^{\sin t}) i+\cos t \sin^2 t (e^{\cos t}) j+\sin t\cos te^{\tan t} k) \cdot (\cos t i-\sin t j+\sec^2 t k) dt$
$\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=\int_0^{\pi/4} (\sin t) (\cos t) (e^{\sin t}) -(\cos t) (\sin^3 t) (e^{\cos t})+(\tan t) (e^{\tan t}) dt$
Need to use calculator.
$\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=0.927136$
