$\int_{-1}^1 x^2 ~dx = \frac{2}{3}.$
Now let us substitute $u=x^2$. Then $du=2x ~dx$ then the definite integration becomes from $1$ to $1$, i.e. $0$. I know I need to partition this from $-1$ to $0$ and $0$ to $1$. But my question is when we substitute anything we do not concern ourselves with the fact that what we are substituting is even or odd. Then why do we need to partition this?