Line integral along the curve $\gamma(t)=(4 \cos t, 4 \sin^2 t) $ Let us consider the vector field in the plane:
$$\vec{F}=\left(x \frac{e^{x^2+y^2} - e}{x^2+y^2},y\frac{e^{x^2+y^2} - e}{x^2+y^2}\right)$$
calculate the line integral along the curve defined by:
$$\gamma:
\begin{cases}
x=4 \cos t \\
y=4 \sin^2 t\\
\end{cases}
$$
with $t\in[0,\pi/2]$.
Any suggestions please?

Here is a direct evaluation using this link
$\begin{align}
dx&=-4\sin t \\
dy&=8\cos t \sin t \\
F&=\Big(\frac{2[\exp{(2\cos 4t +14)}-e] \cos t }{7+\cos 4t},\frac{2[\exp{(2\cos 4t +14})-e] \sin^2 t }{7+\cos 4t}\Big)
\end{align}$
Your integral will hence be 
$\begin{align}I&=-\int_0^{\frac{\pi}{2}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=-\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=-\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt+\int_0^{\frac{\pi}{4}}\frac{2[\exp{(2\cos 4t +14})-e] \sin 4t }{7+\cos 4t}dt\\
&=0.
\end{align}$
where at last I have changed the variable for the second integral from $t$ to $\frac{\pi}{2}-t$.
