Trigonometric Integrals

General form

\int {sin}^{m} (x) {cos}^{n} (x) ⅆ x

\begin{matrix} ℐ & = & \int {sin}^{m} (x) {cos}^{2 s + 1} (x) ⅆ x \\ = & \int {sin}^{m} (x) {({cos}^{2} (x))}^{s} cos (x) ⅆ x \\ = & \int {sin}^{m} (x) {(1 - {sin}^{2} (x))}^{s} ⅆ sin (x) \\ = & \int u^{m} {(1 - u^{2})}^{s} ⅆ u \end{matrix}

\begin{matrix} ℐ & = & \int {sin}^{2 r + 1} (x) {cos}^{n} (x) ⅆ x \\ = & \int {({sin}^{2} (x))}^{r} sin (x) {cos}^{n} (x) ⅆ x \\ = & \int {(1 - {cos}^{2} (x))}^{r} {cos}^{n} (x) ⅆ (-cos (x)) \\ = & - \int {(1 - v^{2})}^{r} v^{n} ⅆ v \end{matrix}

\begin{matrix} ℐ & = & \int {sin}^{2 r} (x) {cos}^{2 s} (x) ⅆ x \\ = & \int {({sin}^{2} (x))}^{r} {({cos}^{2} (x))}^{s} ⅆ x \\ = & \int {(\frac{1 - cos (2 x)}{2})}^{r} {(\frac{1 - sin (2 x)}{2})}^{s} ⅆ x \end{matrix}

\begin{matrix} sin (𝒶 x) sin (𝒷 x) & = & \frac{1}{2} cos ((𝒶 - 𝒷) x) - \frac{1}{2} cos ((𝒶 + 𝒷) x) \\ sin (𝒶 x) cos (𝒷 x) & = & \frac{1}{2} sin ((𝒶 - 𝒷) x) + \frac{1}{2} sin ((𝒶 + 𝒷) x) \\ cos (𝒶 x) cos (𝒷 x) & = & \frac{1}{2} cos ((𝒶 - 𝒷) x) + \frac{1}{2} cos ((𝒶 + 𝒷) x) \end{matrix}