Integral de $$$\sin{\left(x \right)} \cos^{5}{\left(x \right)}$$$

Halla $$$\int \sin{\left(x \right)} \cos^{5}{\left(x \right)}\, dx$$$.

Sea $$$u=\cos{\left(x \right)}$$$.

Entonces $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (los pasos pueden verse »), y obtenemos que $$$\sin{\left(x \right)} dx = - du$$$.

La integral puede reescribirse como

$${\color{red}{\int{\sin{\left(x \right)} \cos^{5}{\left(x \right)} d x}}} = {\color{red}{\int{\left(- u^{5}\right)d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = u^{5}$$$:

$${\color{red}{\int{\left(- u^{5}\right)d u}}} = {\color{red}{\left(- \int{u^{5} d u}\right)}}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=5$$$:

$$- {\color{red}{\int{u^{5} d u}}}=- {\color{red}{\frac{u^{1 + 5}}{1 + 5}}}=- {\color{red}{\left(\frac{u^{6}}{6}\right)}}$$

Recordemos que $$$u=\cos{\left(x \right)}$$$:

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

Por lo tanto,

$$\int{\sin{\left(x \right)} \cos^{5}{\left(x \right)} d x} = - \frac{\cos^{6}{\left(x \right)}}{6}$$

Añade la constante de integración:

$$\int{\sin{\left(x \right)} \cos^{5}{\left(x \right)} d x} = - \frac{\cos^{6}{\left(x \right)}}{6}+C$$

$$$\int \sin{\left(x \right)} \cos^{5}{\left(x \right)}\, dx = - \frac{\cos^{6}{\left(x \right)}}{6} + C$$$A