Integraali $$$5 y^{2} \cos{\left(x \right)}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int 5 y^{2} \cos{\left(x \right)}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=5 y^{2}$$$ ja $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$${\color{red}{\int{5 y^{2} \cos{\left(x \right)} d x}}} = {\color{red}{\left(5 y^{2} \int{\cos{\left(x \right)} d x}\right)}}$$

Kosinin integraali on $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$5 y^{2} {\color{red}{\int{\cos{\left(x \right)} d x}}} = 5 y^{2} {\color{red}{\sin{\left(x \right)}}}$$

Näin ollen,

$$\int{5 y^{2} \cos{\left(x \right)} d x} = 5 y^{2} \sin{\left(x \right)}$$

Lisää integrointivakio:

$$\int{5 y^{2} \cos{\left(x \right)} d x} = 5 y^{2} \sin{\left(x \right)}+C$$

$$$\int 5 y^{2} \cos{\left(x \right)}\, dx = 5 y^{2} \sin{\left(x \right)} + C$$$A