Integral de $$$5 \cos{\left(x \right)} \cos{\left(5 x \right)}$$$

Encontre $$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx$$$.

Reescreva $$$\cos\left(x \right)\cos\left(5 x \right)$$$ utilizando a fórmula $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ com $$$\alpha=x$$$ e $$$\beta=5 x$$$:

$${\color{red}{\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(x \right)} = 5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}$$$:

$${\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}{2}\right)}}$$

Integre termo a termo:

$$\frac{{\color{red}{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{5 \cos{\left(4 x \right)} d x} + \int{5 \cos{\left(6 x \right)} d x}\right)}}}{2}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=5$$$ e $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\int{5 \cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\left(5 \int{\cos{\left(4 x \right)} d x}\right)}}}{2}$$

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

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

Assim,

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2}$$

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2}$$

A integral do cosseno é $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{8}$$

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

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{\left(4 x\right)}} \right)}}{8}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=5$$$ e $$$f{\left(x \right)} = \cos{\left(6 x \right)}$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{5 \cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\left(5 \int{\cos{\left(6 x \right)} d x}\right)}}}{2}$$

Seja $$$u=6 x$$$.

Então $$$du=\left(6 x\right)^{\prime }dx = 6 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{6}$$$.

Portanto,

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{6}$$$ e $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{2}$$

A integral do cosseno é $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{12}$$

Recorde que $$$u=6 x$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{\left(6 x\right)}} \right)}}{12}$$

Portanto,

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left(6 x \right)}}{12}$$

Simplifique:

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}$$

Adicione a constante de integração:

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}+C$$

$$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24} + C$$$A