GATE 2017 EC (EC2) – Question 36
The values of the iterated integrals
$$\int_0^1\left(\int_0^1\frac{x-y}{(x+y)^3}\,dy\right)dx$$
and
$$\int_0^1\left(\int_0^1\frac{x-y}{(x+y)^3}\,dx\right)dy$$
are
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Show answer and explanation
Correct answer: (C) 0.5 and −0.5, respectively
Explanation
For fixed x>0, an antiderivative with respect to y is $y/(x+y)^2$. Evaluating the inner integral gives $1/(x+1)^2$, whose integral from 0 to 1 is 1/2.
For fixed y>0, an antiderivative with respect to x is $-x/(x+y)^2$. The other inner integral is therefore $-1/(1+y)^2$, and its integral is −1/2.
The singularity at the origin prevents an unqualified exchange of integration order: the integrand is not absolutely integrable there. The ordered results are **0.5 and −0.5, C**.