Integral de $$$\cos{\left(\pi n x \right)}$$$ em relação a $$$x$$$

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

Seja $$$u=\pi n x$$$.

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

Logo,

$${\color{red}{\int{\cos{\left(\pi n x \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{\pi n} d u}}}$$

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

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{\pi n} d u}}} = {\color{red}{\frac{\int{\cos{\left(u \right)} d u}}{\pi n}}}$$

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

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

Recorde que $$$u=\pi n x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{\pi n} = \frac{\sin{\left({\color{red}{\pi n x}} \right)}}{\pi n}$$

Portanto,

$$\int{\cos{\left(\pi n x \right)} d x} = \frac{\sin{\left(\pi n x \right)}}{\pi n}$$

Adicione a constante de integração:

$$\int{\cos{\left(\pi n x \right)} d x} = \frac{\sin{\left(\pi n x \right)}}{\pi n}+C$$

$$$\int \cos{\left(\pi n x \right)}\, dx = \frac{\sin{\left(\pi n x \right)}}{\pi n} + C$$$A