differentiating a 2 dimensional line integral

Questions
,
1

I am relatively new to mathematica and I need for a course to calculate the following derivative.

\dfrac{\partial \int_{L(\alpha,u)} f(x,y,t) d(x,y)}{\partial t}
with L(\alpha,u) = \{ (x,y) | x cos(\alpha) + y sin(\alpha) = u \} for \alpha \in [0,\pi[ and u \in \mathbb{R}
for a certain function f(x,y,t).

The problem is that I don’t know how to set the condition x cos(\alpha) + y sin(\alpha) = u in an integral.

Can someone help me with this?

Thanks in advance

Koen

2

This syntax may give what you want:

With[{f = Function[{x, y, t}, t/(1 + x^2 + y^2)], \[Alpha] = Pi/4, 
  u = 1},
 Integrate[f[x, y, t], 
  Element[{x, y}, 
   ImplicitRegion[x*Cos[\[Alpha]] + y*Sin[\[Alpha]] == u, {x, y}]]]]