Evaluate line integral of vector field

Question

F(x,y)= (2x/y) i+ (1+x²/y²) j along the curve x = cos(t), y = sin(t) and t ∈ [π/4, 3π/4]

Answer 1

The formula for the line integral is:

∫f(r)ds = ∫f(r(t))^Tr'(t)dt

So, we need to take the derivative of the curve, and take the dot product of f(r(t)) and r'(t). We have:

r'(t) = [-sin t, cos t]

f(r(t)) = [2 tan t, 1 + cot²t] = [2 tan t, csc²t]

Therefore, we have:

∫f(r(t))^Tr'(t)dt = ∫_π/4^3π/4 (-2tan(t)sin(t) + csc²(t)cos(t))dt

I will leave the integral calculation up to you.