what is the integral of e^1/t ?

Answer 1

It requires a special function that is defined in terms of another integral.

Some common functions that cannot be integrated are used to define such special functions.

In this case, the special function is Ei(m) = integral ( e^m/m) * dm

Integrating by parts: integral ( f * g') = fg - integral (g * f')

Let f = exp(1/x) and g'=1

Then f' = -exp(1/x)/x^2 and g' = x

then the integral becomes: x* exp(1/x) + integral ( exp(1/x)/x ) dx

Recursively integrating exp(1/x)/x:

Let U = 1/x = x^(-1) ---> 1/U = x

dU = -x^(-2) dx

-x^2 * dU = dx

This sub-integral becomes:

integral ( exp(U) du / U ) = Ei(U) = Ei(1/x)

So the anti-derivative (or indefinite integral) is

xe^(1/x)−Ei(1x)+C