Funktion $$$\csc^{2}{\left(x \right)} + 1$$$ integraali

Määritä $$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx$$$.

Integroi termi kerrallaan:

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

Sovella vakiosääntöä $$$\int c\, dx = c x$$$ käyttäen $$$c=1$$$:

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

Funktion $$$\csc^{2}{\left(x \right)}$$$ integraali on $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:

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

Näin ollen,

$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}+C$$

$$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx = \left(x - \cot{\left(x \right)}\right) + C$$$A