Split the interval of integration at $x\=0$, that is, at the center of symmetry.

In the first integral on the right, make the change of variables $x\=-t$ so that $\mathrm{dx}\=-\mathrm{dt}$ and $x\=-a\Rightarrow t\=a$.

In the first integral on the right, reverse the order of integration, thereby changing the sign of the integral.

In the first integral on the right, replace $f\left(-t\right)$ with $-f\left(t\right)$ since $f$ is an odd function.

In the first integral on the right, $t$ is a "dummy" variable, and can be replaced with any other letter. So, change $t$ to $x$.

Arithmetic.

Load the IntegrationTools package.

Control-drag the definite integral.
Context Panel: Assign to a Name≻$q$ 

Apply the Split command to break the interval of integration at $x\=0$.

Control-drag the first summand .
Context Panel: Assign to a Name≻$L_1$

Apply the Change command to $L_1$, changing the integration variable from $x$ to $-t$.

Assign the result to the name $L_2$.

Apply the Flip command to $L_2$, thereby reversing the direction and limits of integration.

Assign the result to the name $L_3$.

Apply the Change command to $L_3$, changing the integration variable from $t$ to $x$.

Assign the result to the name $L_4$.

Expression palette: Evaluation template
Replace $f\left(-x\right)$ with $-f\left(x\right)$.
(Recall that $f$ is odd, so $f\left(-x\right)\=-f\left(x\right)$.)

Add the modified first summand to the second.

Context Panel: Simplify≻Simplify