Evaluate the following integral: ∫1 0 ∫1 0 (𝒚 / 𝟏+𝒙𝒚) 𝒅𝒙𝒅𝒚

Question

I'd be very greatful iff you could answer this question :)

Answer 1

No problem. Exchange the limits on an integral just changes the sign of the answer to that integral.

 

Let λ(x) = 1 +xy, then dλ = ydx. 

Now

∫₁⁰∫₁⁰ y/(1+xy) dxdy          {Substituting 1 +xy with λ}

= ∫₁⁰∫_1+y¹ 1/λ dλdy            {Note the changes in the limits for λ}

= - ∫₁⁰ln(1+y) dy

= ∫₀¹ln(1+y) dy

=∫₁²lnu du                           {Substituting u = 1 + y and changing the limits}

=u(lnu -1)|₁²                        {∫lnx dx = x(lnx -1) which you can derive via integration by parts}

=2(ln2 -1) - 1(ln1-1)

=2ln2 -1.

 

Enjoy!