Calculate the area beneath the x-axis as negative when it is bounded by the x-axis, the curve being y=xe^(x^(2)) and the lines x=0 and x=(ln 2)^(1/2).
"area beneath the x-axis as negative" is confusing in this context
since the curve lies above the x-axis for your given domain.
Do you simply want the area between the curve and the x-axis for the given domain? IF so, then
area = ∫ xe^(x^(2)) dx from 0 to (ln 2)^(1/2)
= (1/2) e^(x^2) from 0 to (ln 2)^(1/2)
= (1/2)e^(ln2) - (1/2)e^0
= (1/2)(2) - 1/2
= 1/2