x(t) = ( cos(2t), -sin(2t)) for time 0 <= t <= pi/4
traverses an anti-clockwise quarter-circle around the origin from (1,0)
x(t) = (1-t, t) for time 0 <= t <= 1
will traverse a line segment from (1,0) to (0,1). Other parameterizations are possible, e.g.
x(t) = (1-2t,2t) for time 0 <= t <= 1/2.
(partial u(x,y))/(partial x) = x y^2
u(x,y) = x^2 y^2/2 + f(y)
where f(y) is some undetermined function of y. Putting this back into the other equation we get
x^2 y = x^2 y + f'(y)
so f is actually a constant, i.e.
u(x,y) = x^2 y^2/2 + C.
Substituting x=0,y=0 we get C=1, so u(1,1) = 3/2.
ln|-2| - ln|-4| = ln 2 - ln 4 = ln (1/2) = - ln 2.
1/(100+x^3) <= 1/99.
Since the integral of 1/99 from -1 to 1 is clearly equal to 2/99, we are done.