Finding the following definite and indefinite integrals I want to calculate the integral 
$$\int_0^{\frac{\pi}{2}} e ^{ \sin t}\, dt.$$
Can we find a primitive function for $f(t) = e ^{\sin t}$?

No, this one is among the functions that do not admit primitives expressible as a composition of elementary functions.
The definite integral, though, can still be computed, through other means than the Leibniz-Newton theorem. Nevertheless, the result is ugly, since it is expressed with the aid of special functions, so just forget about it.
