Even and odd functions using integrals

Question

If $f: [-r, r] \to\mathbb{R}$ is an even function, show that $g(x) = \cos(nx)$ is an even function and $h(x) = \sin(nx)$ is an odd function. Consider:

$\int_{-r}^{r} f(x)\cos(nx)dx = 2\int_{0}^{r} f(x)\cos(nx)dx$

and $\int_{-r}^{r} f(x)\sin(nx)dx = 0$

Deduce analogous results considering $f$ as an odd function.

Answer 1

Let $f$ be an odd function then

$\int_{-r}^{r}f(x)\cos(nx)dx=\int_{-r}^{0}f(x)\cos(nx)dx+\int_{0}^{r}f(x)\cos(nx)dx$

$=-\int_{-r}^{0}f(-(-x))\cos(-n(-x))(-1)dx+\int_{0}^{r}f(x)\cos(nx)dx$

$=-\int_{-r}^{0}-f(-x)\cos(n(-x))(-1)dx+\int_{0}^{r}f(x)\cos(nx)dx$

$=\int_{r}^{0}f(x)cos(nx)dx+\int_{0}^{r}f(x)\cos(nx)dx$

$-\int_{0}^{r}f(x)\cos(nx)dx+\int_{0}^{r}f(x)\cos(nx)dx=0$.

Similarly,

$\int_{-r}^{r}f(x)\sin(nx)dx=\int_{-r}^{0}f(x)\sin(nx)dx+\int_{0}^{r}f(x)\sin(nx)dx$

$-\int_{-r}^{0}f(-x)\sin(n(-x))(-1)dx+\int_{0}^{r}f(x)\sin(nx)dx$

$2\int_{0}^{r}f(x)\sin(nx)dx$.