Integral de $$$e^{\frac{u}{v}}$$$ em relação a $$$u$$$

Encontre $$$\int e^{\frac{u}{v}}\, du$$$.

Seja $$$w=\frac{u}{v}$$$.

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

A integral torna-se

$${\color{red}{\int{e^{\frac{u}{v}} d u}}} = {\color{red}{\int{v e^{w} d w}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ usando $$$c=v$$$ e $$$f{\left(w \right)} = e^{w}$$$:

$${\color{red}{\int{v e^{w} d w}}} = {\color{red}{v \int{e^{w} d w}}}$$

A integral da função exponencial é $$$\int{e^{w} d w} = e^{w}$$$:

$$v {\color{red}{\int{e^{w} d w}}} = v {\color{red}{e^{w}}}$$

Recorde que $$$w=\frac{u}{v}$$$:

$$v e^{{\color{red}{w}}} = v e^{{\color{red}{\frac{u}{v}}}}$$

Portanto,

$$\int{e^{\frac{u}{v}} d u} = v e^{\frac{u}{v}}$$

Adicione a constante de integração:

$$\int{e^{\frac{u}{v}} d u} = v e^{\frac{u}{v}}+C$$

$$$\int e^{\frac{u}{v}}\, du = v e^{\frac{u}{v}} + C$$$A