# Where is symmetry used in the rewriting of the following integral?

Let $$F$$ be a CDF that is symmetric about $$t^* \in T$$.

Consider the following integral:

$$\int_{t^*-\overline{\delta}}^{t^*+\overline{\delta}}cos(t)f(t)dt$$

Using change of variable and the symmetry of $$F$$ we can re-write this integral as

$$\int_{t^*-\overline{\delta}}^{t^*+\overline{\delta}}cos(t)f(t)dt =\int_{-\overline{\delta}}^{\overline{\delta}} cos(t^*+\delta)f(t^*+\delta)d\delta = \int_{0}^{\overline{\delta}} \left(cos(t^*+\delta) + cos(t^*-\delta)\right)f(t^*+\delta)d\delta$$

How is Symmetry being used in the above re-writing of the integral?

I believe its in the second equality (that is changing the bottom limit of integration from $$-\overline{\delta}$$ to $$0$$), but it's not clear to me exactly how.

Is the integral just being split and then "flipped" (non-technical term, sorry) using symmetry? i.e. something like $$\int_{-\overline{\delta}}^{\overline{\delta}} (stuff) = \int_{-\overline{\delta}}^0 (stuff) + \int_0^{\overline{\delta}}$$ and then use symmetry to rewrite $$\int_{-\overline{\delta}}^0 (stuff)$$ as $$\int_0^{\overline{\delta}} ( other stuff)$$?