Integral de $$$\cos^{3}{\left(2 x \right)}$$$

Encontre $$$\int \cos^{3}{\left(2 x \right)}\, dx$$$.

Seja $$$u=2 x$$$.

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

A integral torna-se

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

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

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

Separe um cosseno e escreva o restante em termos de seno, usando a fórmula $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ com $$$\alpha= u $$$:

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

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

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

Assim,

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dv = c v$$$ usando $$$c=1$$$:

$$- \frac{\int{v^{2} d v}}{2} + \frac{{\color{red}{\int{1 d v}}}}{2} = - \frac{\int{v^{2} d v}}{2} + \frac{{\color{red}{v}}}{2}$$

Aplique a regra da potência $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

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

$$\frac{{\color{red}{v}}}{2} - \frac{{\color{red}{v}}^{3}}{6} = \frac{{\color{red}{\sin{\left(u \right)}}}}{2} - \frac{{\color{red}{\sin{\left(u \right)}}}^{3}}{6}$$

Recorde que $$$u=2 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{2} - \frac{\sin^{3}{\left({\color{red}{u}} \right)}}{6} = \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{2} - \frac{\sin^{3}{\left({\color{red}{\left(2 x\right)}} \right)}}{6}$$

Portanto,

$$\int{\cos^{3}{\left(2 x \right)} d x} = - \frac{\sin^{3}{\left(2 x \right)}}{6} + \frac{\sin{\left(2 x \right)}}{2}$$

Simplifique:

$$\int{\cos^{3}{\left(2 x \right)} d x} = \frac{\left(3 - \sin^{2}{\left(2 x \right)}\right) \sin{\left(2 x \right)}}{6}$$

Adicione a constante de integração:

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

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