Solution
= Solution
Parametrize the <straight line> by $r(t)=(t,t,t)$, $0\leq t\leq1$, with $r'(t)=(1,1,1)$. Substituting into the <vector field> and taking the <dot product> gives
$$
v(r(t))\cdot r'(t)=(t+1)e^t+5t^4.
$$
Hence the direct <line integral> is
$$
\boxed{\int_Cv\cdot dr=\int_0^1\bigl[(t+1)e^t+5t^4\bigr]\,dt=\left[te^t+t^5\right]_0^1=e+1.}
$$