Funktion $$$\tan^{2}{\left(x \right)}$$$ integraali

Määritä $$$\int \tan^{2}{\left(x \right)}\, dx$$$.

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

Tällöin $$$x=\operatorname{atan}{\left(u \right)}$$$ ja $$$dx=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du = \frac{du}{u^{2} + 1}$$$ (vaiheet ovat nähtävissä »).

Integraali muuttuu muotoon

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

Kirjoita murtolauseke uudelleen ja jaa se osamurtoihin:

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

Integroi termi kerrallaan:

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

Sovella vakiosääntöä $$$\int c\, du = c u$$$ käyttäen $$$c=1$$$:

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

Funktion $$$\frac{1}{u^{2} + 1}$$$ integraali on $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

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

Muista, että $$$u=\tan{\left(x \right)}$$$:

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

Näin ollen,

$$\int{\tan^{2}{\left(x \right)} d x} = \tan{\left(x \right)} - \operatorname{atan}{\left(\tan{\left(x \right)} \right)}$$

Sievennä:

$$\int{\tan^{2}{\left(x \right)} d x} = - x + \tan{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\tan^{2}{\left(x \right)} d x} = - x + \tan{\left(x \right)}+C$$

$$$\int \tan^{2}{\left(x \right)}\, dx = \left(- x + \tan{\left(x \right)}\right) + C$$$A