Integralen av $$$\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}$$$

Bestäm $$$\int \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Förenkla integranden:

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

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

Skriv om integranden i termer av kosekanten:

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

Integralen av $$$\csc^{2}{\left(x \right)}$$$ är $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:

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

Alltså,

$$\int{\frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)}} d x} = - x - \cot{\left(x \right)}$$

Lägg till integrationskonstanten:

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

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