Integralen av $$$\cot^{4}{\left(2 x \right)}$$$

Bestäm $$$\int \cot^{4}{\left(2 x \right)}\, dx$$$.

Låt $$$u=2 x$$$ vara.

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

Alltså,

$${\color{red}{\int{\cot^{4}{\left(2 x \right)} d x}}} = {\color{red}{\int{\frac{\cot^{4}{\left(u \right)}}{2} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \cot^{4}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cot^{4}{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\cot^{4}{\left(u \right)} d u}}{2}\right)}}$$

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

Då $$$dv=\left(\cot{\left(u \right)}\right)^{\prime }du = - \csc^{2}{\left(u \right)} du$$$ (stegen kan ses »), och vi har att $$$\csc^{2}{\left(u \right)} du = - dv$$$.

Alltså,

$$\frac{{\color{red}{\int{\cot^{4}{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\int{\left(- \frac{v^{4}}{v^{2} + 1}\right)d v}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=-1$$$ och $$$f{\left(v \right)} = \frac{v^{4}}{v^{2} + 1}$$$:

$$\frac{{\color{red}{\int{\left(- \frac{v^{4}}{v^{2} + 1}\right)d v}}}}{2} = \frac{{\color{red}{\left(- \int{\frac{v^{4}}{v^{2} + 1} d v}\right)}}}{2}$$

Eftersom graden hos täljaren inte är mindre än graden hos nämnaren, utför polynomdivision (stegen kan ses »):

$$- \frac{{\color{red}{\int{\frac{v^{4}}{v^{2} + 1} d v}}}}{2} = - \frac{{\color{red}{\int{\left(v^{2} - 1 + \frac{1}{v^{2} + 1}\right)d v}}}}{2}$$

Integrera termvis:

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

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

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

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

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

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

$$- \frac{v^{3}}{6} + \frac{v}{2} - \frac{{\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}}{2} = - \frac{v^{3}}{6} + \frac{v}{2} - \frac{{\color{red}{\operatorname{atan}{\left(v \right)}}}}{2}$$

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

$$- \frac{\operatorname{atan}{\left({\color{red}{v}} \right)}}{2} + \frac{{\color{red}{v}}}{2} - \frac{{\color{red}{v}}^{3}}{6} = - \frac{\operatorname{atan}{\left({\color{red}{\cot{\left(u \right)}}} \right)}}{2} + \frac{{\color{red}{\cot{\left(u \right)}}}}{2} - \frac{{\color{red}{\cot{\left(u \right)}}}^{3}}{6}$$

Kom ihåg att $$$u=2 x$$$:

$$\frac{\cot{\left({\color{red}{u}} \right)}}{2} - \frac{\cot^{3}{\left({\color{red}{u}} \right)}}{6} - \frac{\operatorname{atan}{\left(\cot{\left({\color{red}{u}} \right)} \right)}}{2} = \frac{\cot{\left({\color{red}{\left(2 x\right)}} \right)}}{2} - \frac{\cot^{3}{\left({\color{red}{\left(2 x\right)}} \right)}}{6} - \frac{\operatorname{atan}{\left(\cot{\left({\color{red}{\left(2 x\right)}} \right)} \right)}}{2}$$

Alltså,

$$\int{\cot^{4}{\left(2 x \right)} d x} = - \frac{\cot^{3}{\left(2 x \right)}}{6} + \frac{\cot{\left(2 x \right)}}{2} - \frac{\operatorname{atan}{\left(\cot{\left(2 x \right)} \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{\cot^{4}{\left(2 x \right)} d x} = - \frac{\cot^{3}{\left(2 x \right)}}{6} + \frac{\cot{\left(2 x \right)}}{2} - \frac{\operatorname{atan}{\left(\cot{\left(2 x \right)} \right)}}{2}+C$$

$$$\int \cot^{4}{\left(2 x \right)}\, dx = \left(- \frac{\cot^{3}{\left(2 x \right)}}{6} + \frac{\cot{\left(2 x \right)}}{2} - \frac{\operatorname{atan}{\left(\cot{\left(2 x \right)} \right)}}{2}\right) + C$$$A