Integral de $$$- \sin{\left(3 a \right)}$$$

Encontre $$$\int \left(- \sin{\left(3 a \right)}\right)\, da$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ usando $$$c=-1$$$ e $$$f{\left(a \right)} = \sin{\left(3 a \right)}$$$:

$${\color{red}{\int{\left(- \sin{\left(3 a \right)}\right)d a}}} = {\color{red}{\left(- \int{\sin{\left(3 a \right)} d a}\right)}}$$

Seja $$$u=3 a$$$.

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

Portanto,

$$- {\color{red}{\int{\sin{\left(3 a \right)} d a}}} = - {\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}$$

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

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

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

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

Recorde que $$$u=3 a$$$:

$$\frac{\cos{\left({\color{red}{u}} \right)}}{3} = \frac{\cos{\left({\color{red}{\left(3 a\right)}} \right)}}{3}$$

Portanto,

$$\int{\left(- \sin{\left(3 a \right)}\right)d a} = \frac{\cos{\left(3 a \right)}}{3}$$

Adicione a constante de integração:

$$\int{\left(- \sin{\left(3 a \right)}\right)d a} = \frac{\cos{\left(3 a \right)}}{3}+C$$

$$$\int \left(- \sin{\left(3 a \right)}\right)\, da = \frac{\cos{\left(3 a \right)}}{3} + C$$$A