What is the result of this integral?

Question

$$\int{e^t\over 1+t^2}$$

I don't understand the result of this integral which is supposed to be

Answer 1

Split your answer into two parts, each containing one of the Exponential Integral functions. Then rewrite the functions as integrals, simplify, and combine the integrals again. Here is how to initially set up one of the functions. $$-\frac{ie^i}{2}\text{Ei}(t-i) $$ $$\int_{i-t}^{\infty} \frac{ie^i}{2te^t}$$ This looks messy, but just make sure to your integral limits are the same for both integrals and cancelations should occur when the integrals are combined (I'm on mobile, but if you need more help and no one else provides it I can help more later)