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Integral de $$$\cos{\left(n x \right)}$$$ con respecto a $$$x$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \cos{\left(n x \right)}\, dx$$$.
Solución
Sea $$$u=n x$$$.
Entonces $$$du=\left(n x\right)^{\prime }dx = n dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{n}$$$.
La integral se convierte en
$${\color{red}{\int{\cos{\left(n x \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{n} d u}}}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{n}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:
$${\color{red}{\int{\frac{\cos{\left(u \right)}}{n} d u}}} = {\color{red}{\frac{\int{\cos{\left(u \right)} d u}}{n}}}$$
La integral del coseno es $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:
$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{n} = \frac{{\color{red}{\sin{\left(u \right)}}}}{n}$$
Recordemos que $$$u=n x$$$:
$$\frac{\sin{\left({\color{red}{u}} \right)}}{n} = \frac{\sin{\left({\color{red}{n x}} \right)}}{n}$$
Por lo tanto,
$$\int{\cos{\left(n x \right)} d x} = \frac{\sin{\left(n x \right)}}{n}$$
Añade la constante de integración:
$$\int{\cos{\left(n x \right)} d x} = \frac{\sin{\left(n x \right)}}{n}+C$$
Respuesta
$$$\int \cos{\left(n x \right)}\, dx = \frac{\sin{\left(n x \right)}}{n} + C$$$A