Integralen av $$$\frac{1}{2 - \cos{\left(2 x \right)}}$$$

Bestäm $$$\int \frac{1}{2 - \cos{\left(2 x \right)}}\, dx$$$.

Låt $$$u=2 x$$$ vara.

Då $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{2}$$$.

Integralen kan omskrivas som

$${\color{red}{\int{\frac{1}{2 - \cos{\left(2 x \right)}} d x}}} = {\color{red}{\int{\left(- \frac{1}{2 \left(\cos{\left(u \right)} - 2\right)}\right)d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=- \frac{1}{2}$$$ och $$$f{\left(u \right)} = \frac{1}{\cos{\left(u \right)} - 2}$$$:

$${\color{red}{\int{\left(- \frac{1}{2 \left(\cos{\left(u \right)} - 2\right)}\right)d u}}} = {\color{red}{\left(- \frac{\int{\frac{1}{\cos{\left(u \right)} - 2} d u}}{2}\right)}}$$

Skriv om integranden med hjälp av formeln $$$\cos{\left( u \right)}=\frac{1 - \tan^{2}{\left(\frac{ u }{2} \right)}}{\tan^{2}{\left(\frac{ u }{2} \right)} + 1}$$$:

$$- \frac{{\color{red}{\int{\frac{1}{\cos{\left(u \right)} - 2} d u}}}}{2} = - \frac{{\color{red}{\int{\frac{1}{\frac{1 - \tan^{2}{\left(\frac{u}{2} \right)}}{\tan^{2}{\left(\frac{u}{2} \right)} + 1} - 2} d u}}}}{2}$$

Låt $$$v=\tan{\left(\frac{u}{2} \right)}$$$ vara.

Då gäller $$$u=2 \operatorname{atan}{\left(v \right)}$$$ och $$$du=\left(2 \operatorname{atan}{\left(v \right)}\right)^{\prime }dv = \frac{2}{v^{2} + 1} dv$$$ (stegen kan ses »).

Integralen blir

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

Förenkla:

$$- \frac{{\color{red}{\int{\frac{2}{\left(v^{2} + 1\right) \left(\frac{1 - v^{2}}{v^{2} + 1} - 2\right)} d v}}}}{2} = - \frac{{\color{red}{\int{\left(- \frac{2}{3 v^{2} + 1}\right)d v}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=-2$$$ och $$$f{\left(v \right)} = \frac{1}{3 v^{2} + 1}$$$:

$$- \frac{{\color{red}{\int{\left(- \frac{2}{3 v^{2} + 1}\right)d v}}}}{2} = - \frac{{\color{red}{\left(- 2 \int{\frac{1}{3 v^{2} + 1} d v}\right)}}}{2}$$

Låt $$$w=\sqrt{3} v$$$ vara.

Då $$$dw=\left(\sqrt{3} v\right)^{\prime }dv = \sqrt{3} dv$$$ (stegen kan ses »), och vi har att $$$dv = \frac{\sqrt{3} dw}{3}$$$.

Alltså,

$${\color{red}{\int{\frac{1}{3 v^{2} + 1} d v}}} = {\color{red}{\int{\frac{\sqrt{3}}{3 \left(w^{2} + 1\right)} d w}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ med $$$c=\frac{\sqrt{3}}{3}$$$ och $$$f{\left(w \right)} = \frac{1}{w^{2} + 1}$$$:

$${\color{red}{\int{\frac{\sqrt{3}}{3 \left(w^{2} + 1\right)} d w}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\frac{1}{w^{2} + 1} d w}}{3}\right)}}$$

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

$$\frac{\sqrt{3} {\color{red}{\int{\frac{1}{w^{2} + 1} d w}}}}{3} = \frac{\sqrt{3} {\color{red}{\operatorname{atan}{\left(w \right)}}}}{3}$$

Kom ihåg att $$$w=\sqrt{3} v$$$:

$$\frac{\sqrt{3} \operatorname{atan}{\left({\color{red}{w}} \right)}}{3} = \frac{\sqrt{3} \operatorname{atan}{\left({\color{red}{\sqrt{3} v}} \right)}}{3}$$

Kom ihåg att $$$v=\tan{\left(\frac{u}{2} \right)}$$$:

$$\frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} {\color{red}{v}} \right)}}{3} = \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} {\color{red}{\tan{\left(\frac{u}{2} \right)}}} \right)}}{3}$$

Kom ihåg att $$$u=2 x$$$:

$$\frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{{\color{red}{u}}}{2} \right)} \right)}}{3} = \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{{\color{red}{\left(2 x\right)}}}{2} \right)} \right)}}{3}$$

Alltså,

$$\int{\frac{1}{2 - \cos{\left(2 x \right)}} d x} = \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} \tan{\left(x \right)} \right)}}{3}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{2 - \cos{\left(2 x \right)}} d x} = \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} \tan{\left(x \right)} \right)}}{3}+C$$

$$$\int \frac{1}{2 - \cos{\left(2 x \right)}}\, dx = \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} \tan{\left(x \right)} \right)}}{3} + C$$$A