find the area from 0 to e/λ of g(x)=λx/e-ln(λx)

The result is indeed e/(2lambda):

Definite integral:

integral_0^(e/lambda) ((lambda x)/e-log(lambda x)) dx = e/(2 lambda)

Indefinite integral:

integral ((lambda x)/e-log(lambda x)) dx = (lambda x^2)/(2 e) -xlog(lambda x)+x+constant

I've found the same result but a friend says its incorrect cause it should be (e-2)/2λ

I can't find why it is -2 though

I'm sure your friend is wrong. The answers I gave you are the CORRECT answers, but unfortunately I can't give you step-by-step breakdown of the answer because my software in not working properly.

ok,thanks a lot