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6 1 $begingroup$ Hint given: Write the integrand as an integral. I'm supposed to do this as double integration. My attempt: $$int_0^2 [tan^{-1}y]^{pi x}_{x}$$ $$= int_0^2 int_x^{pi x} frac { mathrm{d}y mathrm{d}x} {y^2+1}$$ $$= int_2^{2pi} int_{frac{y}{pi}}^2 frac { mathrm{d}x mathrm{d}y} {y^2+1}$$ $$= int_2^{2pi} frac { [x]^2_{frac{y}{pi}} mathrm{d}y } { y^2+1}$$ $$= int_2^{2pi} frac { 2- {frac{y}{pi}} mathrm{d}y } { y^2+1}$$ Carrying out this integration, I got, $$2[tan^{-1} 2 pi - tan^{-1} 2] - frac {1}{2 pi} [ln frac {1+4 {pi}^2} {5}]$$ But I'm supposed to get $$2[tan^{-1} 2 pi - tan^{-1} 2] - frac {1}{2 pi} [ln frac {1+4 {pi}^2} {5}]+ [frac {pi-1}{2 pi}] ln 5$$ Can someone please explain where I'm wrong? I've failed to pinpoint my mistake. Thank you.