Complex integral with imaginary exponent: $\int_0^\pi i \exp((i\theta)^{1+i}) d\theta$

Question

How to approach the integral $$ \int_0^\pi i e^{(i\theta)^{1+i}} d\theta $$

I know I can't multiply the exponents, but what can I do?

Am I at least right that the above is equivalent to $\int_0^\pi e^{(i\theta)^{i}} ie^{i\theta} d\theta $? I'm trying to find the integral of $f(z)=z^i$ on the top half of the unit circle.

Answer 1

Using the fact that $~n!=\displaystyle\int_0^\infty\exp\Big(-\sqrt[^{\Large n}]x\Big)~dx,~$ your integral can be expressed in terms of the incomplete $\Gamma$ function.