Integral de $$$\left(u + v\right)^{c - 1}$$$ em relação a $$$u$$$

Encontre $$$\int \left(u + v\right)^{c - 1}\, du$$$.

Seja $$$w=u + v$$$.

Então $$$dw=\left(u + v\right)^{\prime }du = 1 du$$$ (veja os passos »), e obtemos $$$du = dw$$$.

Logo,

$${\color{red}{\int{\left(u + v\right)^{c - 1} d u}}} = {\color{red}{\int{w^{c - 1} d w}}}$$

Aplique a regra da potência $$$\int w^{n}\, dw = \frac{w^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=c - 1$$$:

$${\color{red}{\int{w^{c - 1} d w}}}={\color{red}{\frac{w^{\left(c - 1\right) + 1}}{\left(c - 1\right) + 1}}}={\color{red}{\frac{w^{c}}{c}}}$$

Recorde que $$$w=u + v$$$:

$$\frac{{\color{red}{w}}^{c}}{c} = \frac{{\color{red}{\left(u + v\right)}}^{c}}{c}$$

Portanto,

$$\int{\left(u + v\right)^{c - 1} d u} = \frac{\left(u + v\right)^{c}}{c}$$

Adicione a constante de integração:

$$\int{\left(u + v\right)^{c - 1} d u} = \frac{\left(u + v\right)^{c}}{c}+C$$

$$$\int \left(u + v\right)^{c - 1}\, du = \frac{\left(u + v\right)^{c}}{c} + C$$$A