Complex Conjugate of Integral
I want to know that the equality $$ \overline{\int_{\mathbb R} f(x)dx} = \int_{\mathbb R} \overline{f(x)}dx $$ holds, if the both integral converges. Here $f:\mathbb R \ni x \mapsto f(x)\in \mathbb C $.
Let $f=a+ib$. If the integrals below exist then
$$ \int_{\mathbb R} f(x)dx = \int_{\mathbb R} a(x) \ dx+i\int_{\mathbb R} b(x) \ dx $$
and
$$ \overline {\int_{\mathbb R} f(x)dx}= \int_{\mathbb R} a(x) \ dx-i\int_{\mathbb R} b(x) \ dx =\int_{\mathbb R} a(x)-ib(x)\ dx=\int_{\mathbb R} \overline{f(x)}\ dx.$$