Integral de $$$16 \cos{\left(x \right)} \tan{\left(x \right)}$$$

Encontre $$$\int 16 \cos{\left(x \right)} \tan{\left(x \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=16$$$ e $$$f{\left(x \right)} = \cos{\left(x \right)} \tan{\left(x \right)}$$$:

$${\color{red}{\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x}}} = {\color{red}{\left(16 \int{\cos{\left(x \right)} \tan{\left(x \right)} d x}\right)}}$$

Simplifique o integrando:

$$16 {\color{red}{\int{\cos{\left(x \right)} \tan{\left(x \right)} d x}}} = 16 {\color{red}{\int{\sin{\left(x \right)} d x}}}$$

A integral do seno é $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$16 {\color{red}{\int{\sin{\left(x \right)} d x}}} = 16 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Portanto,

$$\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x} = - 16 \cos{\left(x \right)}$$

Adicione a constante de integração:

$$\int{16 \cos{\left(x \right)} \tan{\left(x \right)} d x} = - 16 \cos{\left(x \right)}+C$$

$$$\int 16 \cos{\left(x \right)} \tan{\left(x \right)}\, dx = - 16 \cos{\left(x \right)} + C$$$A