Find:

Solution (a) If $f(x)={{x^{7}-4x^{3}+x,}}$ then $f({-}x)=({-}x)^{7}-4({-}x)^{3}+({-}x)={-}(x^{7}-4x^{3}+x)={-}f(x).$ Since $f$ is an odd function, $$\int_{-3}^{3}{(x^{7}-4x^{3}+x)}\,dx=0$$

(b) If $g(x) =x^{4}-x^{2}+3,$ then $g(-x) =(-x) ^{4}-(-x) ^{2}+3=x^{4}-x^{2}+3=g(x).$ Since $g$ is an even function, $$\begin{eqnarray*} \int_{-2}^{2}( x^{4}-x^{2}+3) dx &=& 2\int_{0}^{2}(x^{4}-x^{2}+3)\, dx=2\left[ \dfrac{x^{5}}{5}-\dfrac{x^{3}}{3}+3x\right]_{0}^{2} \\ &=& 2\left[ \dfrac{32}{5}-\dfrac{8}{3}+6\right] =\dfrac{292}{15} \end{eqnarray*}$$