The integral ∫etan−1⁡x1+x+x21+x2dx is equal to

Putting tan−1⁡x=u, we have dx1+x2=du∫etan−1⁡x1+x+x21+x2dx=∫eu1+tan⁡u+tan2⁡udu=∫eusec2⁡u+tan⁡udu=tan⁡ueu+C=xetan−1⁡x+C. (using ∫exf(x)+f′(x)dx=exf(x)+C )