Evaluate the line integral $\int_C \mathbf F \cdot \, \mathrm d \mathbf r$ 2

Question

Evaluate the line integral $\int_C \mathbf F \cdot \, \mathrm d \mathbf r$ where, $\mathbf F = (1+xy)e^{xy} \mathbf i+ x^2e^{xy}\mathbf j$ and C is parameterised by $\mathbf r(t)= \cos t \mathbf i+\ 2sin t \mathbf j , t\in[0,\frac\pi2]$

Using $$\int_C \mathbf F \cdot \, \mathrm d \mathbf r = \int^b_a \mathbf F (\mathbf r(t))\mathbf r'(t) dt $$

I have begun to answer the question, this is my current unfinished solution:

$$=\int^\frac\pi2_0 ((1+2costsint)e^{2costsint}\mathbf i + \cos^{2}te^{2costsint}\mathbf j )(\cos t\mathbf i + \sin t\mathbf j) dt$$

$$=\int^\frac\pi2_0e^{sin2t}cost(1+sin2t)+2e^{sin2t}cos^2tsint dt$$

Assuming i'm on the right track, I am not sure how to go about integrating this equation.

Answer 1

The point of this problem is to notice that there is a potential function $\phi$ for the vector field $F$, namely $$ \phi(x,y)=xe^{xy} $$ Then you may use the fundamental theorem of line integrals and avoid any horrific integrals to find $$ \int_{C}F\cdot \mathrm dr=\phi(0,2)-\phi(1,0)=-1 $$

Answer 2

Your $dr$is problematic. You forgot to take derivative of $r$.Note that your field is conservative and the line integral is independent of the path.You may pick a different path or find the potential function to evaluate your integral using the Fundamental Theorem of Line Integrals.