Integraali $$$9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=9 i n t$$$ ja $$$f{\left(x \right)} = \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}$$$:

$${\color{red}{\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(9 i n t \int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$

Kirjoita integroituva uudelleen:

$$9 i n t {\color{red}{\int{\sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}}$$

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

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

Näin ollen,

$$9 i n t {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = 9 i n t {\color{red}{\int{\frac{1}{u} d u}}}$$

Funktion $$$\frac{1}{u}$$$ integraali on $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$9 i n t {\color{red}{\int{\frac{1}{u} d u}}} = 9 i n t {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

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

$$9 i n t \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 9 i n t \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$

Näin ollen,

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Lisää integrointivakio:

$$\int{9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)} d x} = 9 i n t \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}+C$$

$$$\int 9 i n t \sin{\left(x \right)} \cot{\left(x \right)} \csc{\left(x \right)}\, dx = 9 i n t \ln\left(\left|{\sin{\left(x \right)}}\right|\right) + C$$$A