Funktion $$$\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$$ integraali

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

Kirjoita uudelleen kotangentin avulla:

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

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

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

Näin ollen,

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

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

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

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 $$$u=\cot{\left(x \right)}$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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