Integral de $$$\frac{1}{\cos{\left(3 x \right)} + 1}$$$

Encontre $$$\int \frac{1}{\cos{\left(3 x \right)} + 1}\, dx$$$.

Seja $$$u=3 x$$$.

Então $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{3}$$$.

A integral pode ser reescrita como

$${\color{red}{\int{\frac{1}{\cos{\left(3 x \right)} + 1} d x}}} = {\color{red}{\int{\frac{1}{3 \left(\cos{\left(u \right)} + 1\right)} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{3}$$$ e $$$f{\left(u \right)} = \frac{1}{\cos{\left(u \right)} + 1}$$$:

$${\color{red}{\int{\frac{1}{3 \left(\cos{\left(u \right)} + 1\right)} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos{\left(u \right)} + 1} d u}}{3}\right)}}$$

Reescreva o cosseno usando a fórmula do ângulo duplo $$$\cos\left( u \right)=2\cos^2\left(\frac{ u }{2}\right)-1$$$ e simplifique:

$$\frac{{\color{red}{\int{\frac{1}{\cos{\left(u \right)} + 1} d u}}}}{3} = \frac{{\color{red}{\int{\frac{1}{2 \cos^{2}{\left(\frac{u}{2} \right)}} d u}}}}{3}$$

Seja $$$v=\frac{u}{2}$$$.

Então $$$dv=\left(\frac{u}{2}\right)^{\prime }du = \frac{du}{2}$$$ (veja os passos »), e obtemos $$$du = 2 dv$$$.

Portanto,

$$\frac{{\color{red}{\int{\frac{1}{2 \cos^{2}{\left(\frac{u}{2} \right)}} d u}}}}{3} = \frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(v \right)}} d v}}}}{3}$$

Reescreva o integrando em termos da secante:

$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(v \right)}} d v}}}}{3} = \frac{{\color{red}{\int{\sec^{2}{\left(v \right)} d v}}}}{3}$$

A integral de $$$\sec^{2}{\left(v \right)}$$$ é $$$\int{\sec^{2}{\left(v \right)} d v} = \tan{\left(v \right)}$$$:

$$\frac{{\color{red}{\int{\sec^{2}{\left(v \right)} d v}}}}{3} = \frac{{\color{red}{\tan{\left(v \right)}}}}{3}$$

Recorde que $$$v=\frac{u}{2}$$$:

$$\frac{\tan{\left({\color{red}{v}} \right)}}{3} = \frac{\tan{\left({\color{red}{\left(\frac{u}{2}\right)}} \right)}}{3}$$

Recorde que $$$u=3 x$$$:

$$\frac{\tan{\left(\frac{{\color{red}{u}}}{2} \right)}}{3} = \frac{\tan{\left(\frac{{\color{red}{\left(3 x\right)}}}{2} \right)}}{3}$$

Portanto,

$$\int{\frac{1}{\cos{\left(3 x \right)} + 1} d x} = \frac{\tan{\left(\frac{3 x}{2} \right)}}{3}$$

Adicione a constante de integração:

$$\int{\frac{1}{\cos{\left(3 x \right)} + 1} d x} = \frac{\tan{\left(\frac{3 x}{2} \right)}}{3}+C$$

$$$\int \frac{1}{\cos{\left(3 x \right)} + 1}\, dx = \frac{\tan{\left(\frac{3 x}{2} \right)}}{3} + C$$$A