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

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

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

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

Alltså,

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

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

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, du = c u$$$ med $$$c=1$$$:

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

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

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

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

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

Kom ihåg att $$$u=\tan{\left(x \right)}$$$:

$$\operatorname{atan}{\left({\color{red}{u}} \right)} - {\color{red}{u}} + \frac{{\color{red}{u}}^{3}}{3} = \operatorname{atan}{\left({\color{red}{\tan{\left(x \right)}}} \right)} - {\color{red}{\tan{\left(x \right)}}} + \frac{{\color{red}{\tan{\left(x \right)}}}^{3}}{3}$$

Alltså,

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

Förenkla:

$$\int{\tan^{4}{\left(x \right)} d x} = x + \frac{\tan^{3}{\left(x \right)}}{3} - \tan{\left(x \right)}$$

Lägg till integrationskonstanten:

$$\int{\tan^{4}{\left(x \right)} d x} = x + \frac{\tan^{3}{\left(x \right)}}{3} - \tan{\left(x \right)}+C$$

$$$\int \tan^{4}{\left(x \right)}\, dx = \left(x + \frac{\tan^{3}{\left(x \right)}}{3} - \tan{\left(x \right)}\right) + C$$$A