Integral de $$$\left(2 x - 3\right)^{7}$$$

Halla $$$\int \left(2 x - 3\right)^{7}\, dx$$$.

Sea $$$u=2 x - 3$$$.

Entonces $$$du=\left(2 x - 3\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

Entonces,

$${\color{red}{\int{\left(2 x - 3\right)^{7} d x}}} = {\color{red}{\int{\frac{u^{7}}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = u^{7}$$$:

$${\color{red}{\int{\frac{u^{7}}{2} d u}}} = {\color{red}{\left(\frac{\int{u^{7} d u}}{2}\right)}}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=7$$$:

$$\frac{{\color{red}{\int{u^{7} d u}}}}{2}=\frac{{\color{red}{\frac{u^{1 + 7}}{1 + 7}}}}{2}=\frac{{\color{red}{\left(\frac{u^{8}}{8}\right)}}}{2}$$

Recordemos que $$$u=2 x - 3$$$:

$$\frac{{\color{red}{u}}^{8}}{16} = \frac{{\color{red}{\left(2 x - 3\right)}}^{8}}{16}$$

Por lo tanto,

$$\int{\left(2 x - 3\right)^{7} d x} = \frac{\left(2 x - 3\right)^{8}}{16}$$

Añade la constante de integración:

$$\int{\left(2 x - 3\right)^{7} d x} = \frac{\left(2 x - 3\right)^{8}}{16}+C$$

$$$\int \left(2 x - 3\right)^{7}\, dx = \frac{\left(2 x - 3\right)^{8}}{16} + C$$$A