Funktion $$$\cot^{2}{\left(x + \frac{\pi}{4} \right)}$$$ integraali

Määritä $$$\int \cot^{2}{\left(x + \frac{\pi}{4} \right)}\, dx$$$.

Olkoon $$$u=x + \frac{\pi}{4}$$$.

Tällöin $$$du=\left(x + \frac{\pi}{4}\right)^{\prime }dx = 1 dx$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$dx = du$$$.

Integraali voidaan kirjoittaa muotoon

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

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

Tällöin $$$dv=\left(\cot{\left(u \right)}\right)^{\prime }du = - \csc^{2}{\left(u \right)} du$$$ (vaiheet ovat nähtävissä ») ja saamme, että $$$\csc^{2}{\left(u \right)} du = - dv$$$.

Integraali muuttuu muotoon

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ käyttäen $$$c=-1$$$ ja $$$f{\left(v \right)} = \frac{v^{2}}{v^{2} + 1}$$$:

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

Kirjoita murtolauseke uudelleen ja jaa se osamurtoihin:

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

Integroi termi kerrallaan:

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

Sovella vakiosääntöä $$$\int c\, dv = c v$$$ käyttäen $$$c=1$$$:

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

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

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

Muista, että $$$v=\cot{\left(u \right)}$$$:

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

Muista, että $$$u=x + \frac{\pi}{4}$$$:

$$- \cot{\left({\color{red}{u}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{u}} \right)} \right)} = - \cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} \right)}$$

Näin ollen,

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

Lisää integrointivakio:

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

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