Is :$I(t)= \int_{0}^{t} ( \sin x+\cos x)^{\operatorname{erf}(x)}dx$ complex or real for $t \to \infty $?

Question

let us to check the behavior of this integral :$I(t)= \int_{0}^{t} ( \sin x+\cos x)^{\operatorname{erf}(x)}dx$, Really this integral gives to me for small $t$ values close to $t$ as shown here , and it's values takes negative values and positive with imaginary part close to $0$ , Really I want to ask this question: What is the value of this integral for $t \to \infty $ is it complex values or real ?

Answer 1

$\cos(x)+\sin(x)$ is negative in part of the region of integration, and raising a negative number to a positive real number is not well defined.

More specifically, if $a > 0$, then $(-a)^x =a^x(-1)^x =a^xe^{\pi i x} =a^x(\cos(\pi x)+i\sin(\pi x)) $ and this is where the complex numbers come in.