Evaluating this line integral Evaluate the following line integral
$\int 2x+y \,dx +xy\,dy$ from $(-1,2)$ to $(2,5)$ where $y=x^2 +1$
When I integrated it w.r.t $x$ I found the integral
$$\int  2x^4 +3x^2 +2x+1 \, dx=\frac{141}{5}$$
from $x=-1$ to $x=2$
But when I tried it w.r.t $y$ I got a confusion with evaluating the limits of $y$ since $y=x^2 +1 $ then $x=\sqrt{y-1}$  where $y$ belongs to $[1,5]$
and $x=-\sqrt{y-1}$ where $y$ belongs to $[1,2]$

Your evaluation
$$\int_C (2x+y)dx +xydy=\int_{x=-1}^2 ((2x+x^2+1) +x(x^2+1)(2x))dx=\frac{141}{5}$$
is correct. Using $y$ as a parameter is more difficult, but you should obtain the same result.
\begin{align*}
\int_C (2x+y)dx +xydy&=\int_{y=2}^1 ((2f(y)+y)f'(y) +f(y)y)dy
+\int_{y=1}^5 ((2g(y)+y)g'(y) +g(y)y)dy\\
&=\int_{y=2}^1 \left(1-\frac{y}{2\sqrt{y-1}} -y\sqrt{y-1}\right)dy\\
&\qquad+\int_{y=1}^5 \left(1+\frac{y}{2\sqrt{y-1}} +y\sqrt{y-1}\right)dy
\end{align*}
where $f(y)=-\sqrt{y-1}$ and $g(y)=\sqrt{y-1}$.
Can you take it from here?
