The integral of 5/(x^2 + 1) dx from -1 to 1. (The TheIntegral.jpg attachment shows this in a aesthetically-pleasing way.)
sin^2 (θ) + cos^2 (θ) = 1
1 + tan^2 (θ) = sec^2 (θ)
cot^2 (θ) + 1 = csc^2 (θ)
x = tan(θ)
x = cot(θ)
The Attempt at a Solution
When I am solving the integral, I successfully compute it using the x = tan(θ) substitution but when I use the x = cot(θ) substitution, I get the wrong answer.
My specific work is attached as MyWork.jpg. The work in black is the work I got correct (as indicated by the red writing) and, the work in green is the work I get wrong (as indicated by the red writing).
Could someone please tell me what I am doing wrong when using the cot(θ) substitution?