Answer
$\frac{11e-48}{12e}$
Work Step by Step
$\int_CF.dr=\int_0^{1}2te^{-t}dt+\int_0^{1}-3t^5-t^2-t^3dt$
$=[-2te^{-t}-2e^{-t}]_0^{1}+[-\frac{t^6}{2}-\frac{t^3}{3}-\frac{t^4}{4}]_0^{1}$
$=2-\frac{4}{e}-\frac{13}{12}$
$=\frac{11e-48}{12e}$