Evaluate ∫∫ e^(y/x) dy dx

I = ∫₀¹ ∫₀^{x²} e^(y/x) dy dx

The integral is easy in the given order, because integrating e^(y/x) with respect to y just brings out a factor x.

Inner integral (x constant)

∫₀^{x²} e^(y/x) dy = [ x e^(y/x) ]₀^{x²}
                   = x e^x − x

Outer integral

I = ∫₀¹ (x eˣ − x) dx
  = ∫₀¹ x eˣ dx − ∫₀¹ x dx

By parts: ∫ x eˣ dx = x eˣ − eˣ, so ∫₀¹ x eˣ dx = (e − e) − (0 − 1) = 1.

I = 1 − 1/2 = 1/2

Answer: 1/2.