Integral de $$$\cos^{4}{\left(7 x \right)}$$$

Encontre $$$\int \cos^{4}{\left(7 x \right)}\, dx$$$.

Seja $$$u=7 x$$$.

Então $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{7}$$$.

Logo,

$${\color{red}{\int{\cos^{4}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{7}$$$ e $$$f{\left(u \right)} = \cos^{4}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cos^{4}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\cos^{4}{\left(u \right)} d u}}{7}\right)}}$$

Aplique a fórmula de redução de potência $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$ com $$$\alpha= u $$$:

$$\frac{{\color{red}{\int{\cos^{4}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{8}$$$ e $$$f{\left(u \right)} = 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3$$$:

$$\frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}}{7} = \frac{{\color{red}{\left(\frac{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}{8}\right)}}}{7}$$

Integre termo a termo:

$$\frac{{\color{red}{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}}}{56} = \frac{{\color{red}{\left(\int{3 d u} + \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}}{56}$$

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=3$$$:

$$\frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{3 d u}}}}{56} = \frac{\int{4 \cos{\left(2 u \right)} d u}}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(3 u\right)}}}{56}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=4$$$ e $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$:

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{4 \cos{\left(2 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}}{56}$$

Seja $$$v=2 u$$$.

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

Portanto,

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{14} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{14}$$

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

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{{\color{red}{\sin{\left(v \right)}}}}{28}$$

Recorde que $$$v=2 u$$$:

$$\frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{v}} \right)}}{28} = \frac{3 u}{56} + \frac{\int{\cos{\left(4 u \right)} d u}}{56} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{28}$$

Seja $$$v=4 u$$$.

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

A integral pode ser reescrita como

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56}$$

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

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{56} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{56}$$

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

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{{\color{red}{\sin{\left(v \right)}}}}{224}$$

Recorde que $$$v=4 u$$$:

$$\frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{v}} \right)}}{224} = \frac{3 u}{56} + \frac{\sin{\left(2 u \right)}}{28} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{224}$$

Recorde que $$$u=7 x$$$:

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{224} + \frac{3 {\color{red}{u}}}{56} = \frac{\sin{\left(2 {\color{red}{\left(7 x\right)}} \right)}}{28} + \frac{\sin{\left(4 {\color{red}{\left(7 x\right)}} \right)}}{224} + \frac{3 {\color{red}{\left(7 x\right)}}}{56}$$

Portanto,

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$$

Simplifique:

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}$$

Adicione a constante de integração:

$$\int{\cos^{4}{\left(7 x \right)} d x} = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224}+C$$

$$$\int \cos^{4}{\left(7 x \right)}\, dx = \frac{84 x + 8 \sin{\left(14 x \right)} + \sin{\left(28 x \right)}}{224} + C$$$A