## 2.4 Hyperbolic Functions

### 2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x

2.411 $\begin{array}{lllll}{\displaystyle \int {\text{sinh}}^{p}x{\text{cosh}}^{q}x\text{}\text{d}x}\hfill & =\hfill & \frac{{\text{sinh}}^{p+1}x\text{}{\text{cosh}}^{q-1}x}{p+q}\hfill & +\hfill & \frac{q-1}{p+q}{\displaystyle \int {\text{sinh}}^{p}x\text{}{\text{cosh}}^{q-2}x\text{}\text{d}x}\hfill \\ =\hfill & \frac{{\text{sinh}}^{p-1}x\text{}{\text{cosh}}^{q+1}x}{p+q}\hfill & -\hfill & \frac{p-1}{p+q}{\displaystyle \int {\text{sinh}}^{p-2}x\text{}{\text{cosh}}^{q}x\text{}\text{d}x}\hfill \\ =\hfill & \frac{{\text{sinh}}^{p-1}x\text{}{\text{cosh}}^{q+1}x}{q+1}\hfill & -\hfill & \frac{p-1}{q+1}{\displaystyle \int {\text{sinh}}^{p-2}x\text{}{\text{cosh}}^{q+2}x\text{}\text{d}x}\hfill \\ =\hfill & \frac{{\text{sinh}}^{p+1}x\text{}{\text{cosh}}^{q-1}x}{p+1}\hfill & -\hfill & \frac{q-1}{p+1}{\displaystyle \int {\text{sinh}}^{p+2}x\text{}{\text{cosh}}^{q-2}x\text{}\text{d}x}\hfill \\ =\hfill & \frac{{\text{sinh}}^{p+1}x\text{}{\text{cosh}}^{q+1}x}{p+1}\hfill & -\hfill & \frac{p+q+2}{p+1}{\displaystyle \int {\text{sinh}}^{p+2}x\text{}{\text{cosh}}^{q}x\text{}\text{d}x}\hfill \\ =\hfill & \frac{{\text{sinh}}^{p+1}x\text{}{\text{cosh}}^{q+1}x}{q+1}\hfill & +\hfill & \frac{p+q+2}{q+1}{\displaystyle \int {\text{sinh}}^{p}x\text{}{\text{cosh}}^{q+2}x\text{}\text{d}x}\hfill \end{array}$

2.412

1.

$\begin{array}{lll}{{\displaystyle \int}}^{\text{}}{\text{sinh}}^{p}x{\text{cosh}}^{2n}x\text{d}x\hfill & =\hfill & \frac{{\text{sinh}}^{p+1}}{2n+p}[{\text{cosh}}^{2n-1}x\hfill \\ +{\displaystyle \sum}_{k=1}^{n-1}\frac{(2n-1)(2n-3)\dots (2n-2k+1)}{(2n+p-2)(2n+p-4)\dots (2n+p-2k)}{\text{cosh}}^{2n-2k-1}x]\hfill \\ +\frac{(2n-1)!!}{(2n+p)(2n+p-2)\dots (p-2)}{{\displaystyle \int}}^{\text{}}{\text{sinh}}^{p}x\text{d}x\hfill \end{array}$

This formula is applicable for arbitrary real p except for the following negative even integers: ...

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