Integrating an Even or Odd Function

Find:

1. \(\int_{-3}^{3}{(x^{7}-4x^{3}+x)}\,dx\)
2. \(\int_{-2}^{2}(x^{4}-x^{2}+3)\, dx\)

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, \[ \begin{eqnarray*} \int_{-2}^{2}( x^{4}-x^{2}+3) dx &=& 2\int_{0}^{2}(x^{4}-x^{2}+3)\, dx=2\left[ \dfrac{x^{5}}{5}-\dfrac{x^{3}}{3}+3x\right]_{0}^{2} \\ &=& 2\left[ \dfrac{32}{5}-\dfrac{8}{3}+6\right] =\dfrac{292}{15} \end{eqnarray*} \]