Menu

Nachlesen

Areakosinus hyperbolicus (Integrationsregel)

Herleitung der Stammfunktion von $arcosh (x)$

\begin{aligned} \int arcosh (x) d x & = \int 1 \cdot arcosh (x) d x \\ = x \cdot arcosh (x) - \int x \cdot \frac{1}{\sqrt{x^{2} - 1}} d x \end{aligned}

Substitution von $t = x^{2} - 1$ . Aus $t = x^{2} - 1$ folgt $d x = \frac{d t}{2 x}$ .

\begin{aligned} \int x \cdot \frac{1}{\sqrt{x^{2} - 1}} d x & = \int x \cdot \frac{1}{\sqrt{t}} \cdot \frac{d t}{2 x} \\ = \int \frac{1}{2} \cdot \frac{1}{\sqrt{t}} d t \\ = \int \frac{1}{2} \cdot t^{- \frac{1}{2}} d t \\ = t^{\frac{1}{2}} \\ = \sqrt{x^{2} - 1} \end{aligned}

Es folgt:

\begin{aligned} \int arcosh (x) d x & = x \cdot arcosh (x) - \sqrt{x^{2} - 1} \end{aligned}

Herleitung der Stammfunktion von ${arcosh}^{- 1} (x)$

N/A

Herleitung der Stammfunktion von ${arcosh}^{n} (x)$

Partielle Integration:

\begin{aligned} \int {arcosh}^{n} (x) d x & = \int arcosh (x) \cdot {arcosh}^{n - 1} (x) d x \\ = (x \cdot arcosh (x) - \sqrt{x^{2} - 1}) \cdot {arcosh}^{n - 1} (x) \\ - \int (x \cdot arcosh (x) - \sqrt{x^{2} - 1}) \cdot (n - 1) \cdot {arcosh}^{n - 2} (x) \cdot \frac{1}{\sqrt{x^{2} - 1}} d x \\ = x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) \\ - (n - 1) \cdot (\int \frac{x \cdot arcosh (x) \cdot {arcosh}^{n - 2} (x)}{\sqrt{x^{2} - 1}} d x - \int \frac{\sqrt{x^{2} - 1} \cdot {arcosh}^{n - 2} (x)}{\sqrt{x^{2} - 1}} d x) \\ = x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) \\ - (n - 1) \cdot \int \frac{x \cdot {arcosh}^{n - 1} (x)}{\sqrt{x^{2} - 1}} d x + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \end{aligned}

Umformung von $(n - 1) \cdot \int \frac{x \cdot {arcosh}^{n - 1} (x)}{\sqrt{x^{2} - 1}} d x$ :

\begin{aligned} (n - 1) \cdot \int \frac{x \cdot {arcosh}^{n - 1} (x)}{\sqrt{x^{2} - 1}} d x & = \frac{n - 1}{n} \cdot \int \frac{x \cdot n \cdot {arcosh}^{n - 1} (x)}{\sqrt{x^{2} - 1}} d x \\ = \frac{n - 1}{n} \cdot \int x \cdot \frac{n \cdot {arcosh}^{n - 1} (x)}{\sqrt{x^{2} - 1}} d x \\ = \frac{n - 1}{n} \cdot (x \cdot {arcosh}^{n} (x) - \int 1 \cdot {arcosh}^{n} (x) d x) \end{aligned}

Einsetzen in das Integral $\int {arcosh}^{n} (x) d x$ :

\begin{aligned} \int {arcosh}^{n} (x) d x & = x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) \\ - \frac{n - 1}{n} \cdot (x \cdot {arcosh}^{n} (x) - \int 1 \cdot {arcosh}^{n} (x) d x) + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \\ = x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) \\ - \frac{n - 1}{n} \cdot x \cdot {arcosh}^{n} (x) + \frac{n - 1}{n} \cdot \int {arcosh}^{n} (x) d x + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \\ = (1 - \frac{n - 1}{n}) \cdot x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) \\ + \frac{n - 1}{n} \cdot \int {arcosh}^{n} (x) d x + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \end{aligned}

Umstellen nach $\int {arcosh}^{n} (x) d x$ :

\begin{aligned} (1 - \frac{n - 1}{n}) \cdot \int {arcosh}^{n} (x) d x & = (1 - \frac{n - 1}{n}) \cdot x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \\ \frac{1}{n} \cdot \int {arcosh}^{n} (x) d x & = \frac{1}{n} \cdot x \cdot {arcosh}^{n} (x) - \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) + (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \\ \int {arcosh}^{n} (x) d x & = x \cdot {arcosh}^{n} (x) - n \cdot \sqrt{x^{2} - 1} \cdot {arcosh}^{n - 1} (x) + n \cdot (n - 1) \cdot \int {arcosh}^{n - 2} (x) d x \end{aligned}

Herleitung der Stammfunktion von ${arcosh}^{- n} (x)$

N/A