Q.Evaluate $\displaystyle\int_{-1}^{1} (x^{2} + 1)\,dx$ using the even/odd function property, and state the value of $\displaystyle\int_{-2}^{2} x^{3}\,dx$.

The limits are symmetric about $0$: test each integrand for even/oddness (P7) before integrating.

Both integrals have symmetric limits, so test the integrand's symmetry first (§5).

First integral — $f(x) = x^2 + 1$. Replace $x$ by $-x$: $f(-x) = (-x)^2 + 1 = x^2 + 1 = f(x)$, so $f$ is even. By P7, $\displaystyle\int_{-1}^{1} f(x)\,dx = 2\int_{0}^{1}(x^2 + 1)\,dx$. Now

Second integral — $g(x) = x^3$. Replace $x$ by $-x$: $g(-x) = (-x)^3 = -x^3 = -g(x)$, so $g$ is odd. By P7, $\displaystyle\int_{-2}^{2} x^3\,dx = 0$ — no antiderivative computation needed. …

A common error is to declare $x^2 + 1$ "neither even nor odd" because of the $+1$ — but a constant is even, and a sum of even functions is even, so $x^2+1$ is even. Another is forgetting that the even case gives twice the half-integr …