How do we solve this integral?

Question

Evaluate the following integral using a change of variables.

∫∫((y-x)/(y+2x+1))^4 dA, where R is th parallelogram bounded by y-x=2, y-x=3, y+2x=0, and y+2x=3

R

Answer 1

y -x = u; y + 2x = v; u = 2, u = 3; v = 0 and v = 3;

jacobian J = I - 1 1I = - 3

I 2 1I

∫∫_R((y-x)/(y+2x+1))⁴ dA,= ∫₂³udu∫₀³v⁴·3dv= u²/2₂³·3v⁵/5₀³ = (9/2 - 4/2)·3⁶/5 = 364.5