Integralen av $$$\frac{\sqrt{x^{2} - 100}}{x}$$$

Bestäm $$$\int \frac{\sqrt{x^{2} - 100}}{x}\, dx$$$.

Låt $$$x=10 \cosh{\left(u \right)}$$$ vara.

Då $$$dx=\left(10 \cosh{\left(u \right)}\right)^{\prime }du = 10 \sinh{\left(u \right)} du$$$ (stegen kan ses »).

Det följer också att $$$u=\operatorname{acosh}{\left(\frac{x}{10} \right)}$$$.

Alltså,

$$$\frac{\sqrt{x^{2} - 100}}{x} = \frac{\sqrt{100 \cosh^{2}{\left( u \right)} - 100}}{10 \cosh{\left( u \right)}}$$$

Använd identiteten $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$:

$$$\frac{\sqrt{100 \cosh^{2}{\left( u \right)} - 100}}{10 \cosh{\left( u \right)}}=\frac{\sqrt{\cosh^{2}{\left( u \right)} - 1}}{\cosh{\left( u \right)}}=\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)}}$$$

Om vi antar att $$$\sinh{\left( u \right)} \ge 0$$$, erhåller vi följande:

$$$\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)}} = \frac{\sinh{\left( u \right)}}{\cosh{\left( u \right)}}$$$

Alltså,

$${\color{red}{\int{\frac{\sqrt{x^{2} - 100}}{x} d x}}} = {\color{red}{\int{\frac{10 \sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=10$$$ och $$$f{\left(u \right)} = \frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}}$$$:

$${\color{red}{\int{\frac{10 \sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}} = {\color{red}{\left(10 \int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}\right)}}$$

Multiplicera täljare och nämnare med en hyperbolisk cosinus och skriv allt annat i termer av hyperbolisk sinus med hjälp av formeln $$$\cosh^2\left(\alpha \right)=\sinh^2\left(\alpha \right)+1$$$ med $$$\alpha= u $$$:

$$10 {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}} = 10 {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \cosh{\left(u \right)}}{\sinh^{2}{\left(u \right)} + 1} d u}}}$$

Låt $$$v=\sinh{\left(u \right)}$$$ vara.

Då $$$dv=\left(\sinh{\left(u \right)}\right)^{\prime }du = \cosh{\left(u \right)} du$$$ (stegen kan ses »), och vi har att $$$\cosh{\left(u \right)} du = dv$$$.

Integralen kan omskrivas som

$$10 {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \cosh{\left(u \right)}}{\sinh^{2}{\left(u \right)} + 1} d u}}} = 10 {\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}}$$

Skriv om och dela upp bråket:

$$10 {\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}} = 10 {\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}}$$

Integrera termvis:

$$10 {\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}} = 10 {\color{red}{\left(\int{1 d v} - \int{\frac{1}{v^{2} + 1} d v}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dv = c v$$$ med $$$c=1$$$:

$$- 10 \int{\frac{1}{v^{2} + 1} d v} + 10 {\color{red}{\int{1 d v}}} = - 10 \int{\frac{1}{v^{2} + 1} d v} + 10 {\color{red}{v}}$$

Integralen av $$$\frac{1}{v^{2} + 1}$$$ är $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:

$$10 v - 10 {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}} = 10 v - 10 {\color{red}{\operatorname{atan}{\left(v \right)}}}$$

Kom ihåg att $$$v=\sinh{\left(u \right)}$$$:

$$- 10 \operatorname{atan}{\left({\color{red}{v}} \right)} + 10 {\color{red}{v}} = - 10 \operatorname{atan}{\left({\color{red}{\sinh{\left(u \right)}}} \right)} + 10 {\color{red}{\sinh{\left(u \right)}}}$$

Kom ihåg att $$$u=\operatorname{acosh}{\left(\frac{x}{10} \right)}$$$:

$$10 \sinh{\left({\color{red}{u}} \right)} - 10 \operatorname{atan}{\left(\sinh{\left({\color{red}{u}} \right)} \right)} = 10 \sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{10} \right)}}} \right)} - 10 \operatorname{atan}{\left(\sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{10} \right)}}} \right)} \right)}$$

Alltså,

$$\int{\frac{\sqrt{x^{2} - 100}}{x} d x} = 10 \sqrt{\frac{x}{10} - 1} \sqrt{\frac{x}{10} + 1} - 10 \operatorname{atan}{\left(\sqrt{\frac{x}{10} - 1} \sqrt{\frac{x}{10} + 1} \right)}$$

Förenkla:

$$\int{\frac{\sqrt{x^{2} - 100}}{x} d x} = \sqrt{x - 10} \sqrt{x + 10} - 10 \operatorname{atan}{\left(\frac{\sqrt{x - 10} \sqrt{x + 10}}{10} \right)}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sqrt{x^{2} - 100}}{x} d x} = \sqrt{x - 10} \sqrt{x + 10} - 10 \operatorname{atan}{\left(\frac{\sqrt{x - 10} \sqrt{x + 10}}{10} \right)}+C$$

$$$\int \frac{\sqrt{x^{2} - 100}}{x}\, dx = \left(\sqrt{x - 10} \sqrt{x + 10} - 10 \operatorname{atan}{\left(\frac{\sqrt{x - 10} \sqrt{x + 10}}{10} \right)}\right) + C$$$A