Integral de $$$\tan{\left(4 x \right)}$$$

Encontre $$$\int \tan{\left(4 x \right)}\, dx$$$.

Seja $$$u=4 x$$$.

Então $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{4}$$$.

A integral pode ser reescrita como

$${\color{red}{\int{\tan{\left(4 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)}}{4} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{4}$$$ e $$$f{\left(u \right)} = \tan{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\tan{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{\tan{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} d u}}}}{4}$$

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

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

A integral pode ser reescrita como

$$\frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} d u}}}}{4} = \frac{{\color{red}{\int{\left(- \frac{1}{v}\right)d v}}}}{4}$$

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

$$\frac{{\color{red}{\int{\left(- \frac{1}{v}\right)d v}}}}{4} = \frac{{\color{red}{\left(- \int{\frac{1}{v} d v}\right)}}}{4}$$

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

$$- \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{4} = - \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{4}$$

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

$$- \frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{4} = - \frac{\ln{\left(\left|{{\color{red}{\cos{\left(u \right)}}}}\right| \right)}}{4}$$

Recorde que $$$u=4 x$$$:

$$- \frac{\ln{\left(\left|{\cos{\left({\color{red}{u}} \right)}}\right| \right)}}{4} = - \frac{\ln{\left(\left|{\cos{\left({\color{red}{\left(4 x\right)}} \right)}}\right| \right)}}{4}$$

Portanto,

$$\int{\tan{\left(4 x \right)} d x} = - \frac{\ln{\left(\left|{\cos{\left(4 x \right)}}\right| \right)}}{4}$$

Adicione a constante de integração:

$$\int{\tan{\left(4 x \right)} d x} = - \frac{\ln{\left(\left|{\cos{\left(4 x \right)}}\right| \right)}}{4}+C$$

$$$\int \tan{\left(4 x \right)}\, dx = - \frac{\ln\left(\left|{\cos{\left(4 x \right)}}\right|\right)}{4} + C$$$A