(1)

\(f\left(x\right)\)は奇関数なので\(f\left(x\right)=-f\left(-x\right)\)となり、
\begin{align*} \int_{-a}^{a}f\left(x\right)dx & =\int_{-a}^{0}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =-\int_{-a}^{0}f\left(-x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =\int_{a}^{0}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =-\int_{0}^{a}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =0 \end{align*} となるので与式は成り立つ。

(2)

\(f\left(x\right)\)は偶関数なので\(f\left(x\right)=f\left(-x\right)\)となり、
\begin{align*} \int_{-a}^{a}f\left(x\right)dx & =\int_{-a}^{0}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =\int_{-a}^{0}f\left(-x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =-\int_{a}^{0}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =\int_{0}^{a}f\left(x\right)dx+\int_{0}^{a}f\left(x\right)dx\\ & =2\int_{0}^{a}f\left(x\right)dx \end{align*} となるので与式は成り立つ。