Integral de $$$n \tan{\left(x \right)}$$$ em relação a $$$x$$$

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

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

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

Reescreva a reta tangente como $$$\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}$$$:

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

Seja $$$u=\cos{\left(x \right)}$$$.

Então $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (veja os passos »), e obtemos $$$\sin{\left(x \right)} dx = - du$$$.

Portanto,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-1$$$ e $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$n {\color{red}{\int{\left(- \frac{1}{u}\right)d u}}} = n {\color{red}{\left(- \int{\frac{1}{u} d u}\right)}}$$

A integral de $$$\frac{1}{u}$$$ é $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Recorde que $$$u=\cos{\left(x \right)}$$$:

$$- n \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - n \ln{\left(\left|{{\color{red}{\cos{\left(x \right)}}}}\right| \right)}$$

Portanto,

$$\int{n \tan{\left(x \right)} d x} = - n \ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}$$

Adicione a constante de integração:

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

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