Integral de $$$a^{4^{x}}$$$ em relação a $$$x$$$

Encontre $$$\int a^{4^{x}}\, dx$$$.

Seja $$$u=4^{x}$$$.

Então $$$du=\left(4^{x}\right)^{\prime }dx = 4^{x} \ln{\left(4 \right)} dx$$$ (veja os passos »), e obtemos $$$4^{x} dx = \frac{du}{\ln{\left(4 \right)}}$$$.

Portanto,

$${\color{red}{\int{a^{4^{x}} d x}}} = {\color{red}{\int{\frac{a^{u}}{2 u \ln{\left(2 \right)}} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2 \ln{\left(2 \right)}}$$$ e $$$f{\left(u \right)} = \frac{a^{u}}{u}$$$:

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

Alterar a base:

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

Seja $$$v=u \ln{\left(a \right)}$$$.

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

Assim,

$$\frac{{\color{red}{\int{\frac{e^{u \ln{\left(a \right)}}}{u} d u}}}}{2 \ln{\left(2 \right)}} = \frac{{\color{red}{\int{\frac{e^{v}}{v} d v}}}}{2 \ln{\left(2 \right)}}$$

Esta integral (Integral Exponencial) não possui forma fechada:

$$\frac{{\color{red}{\int{\frac{e^{v}}{v} d v}}}}{2 \ln{\left(2 \right)}} = \frac{{\color{red}{\operatorname{Ei}{\left(v \right)}}}}{2 \ln{\left(2 \right)}}$$

Recorde que $$$v=u \ln{\left(a \right)}$$$:

$$\frac{\operatorname{Ei}{\left({\color{red}{v}} \right)}}{2 \ln{\left(2 \right)}} = \frac{\operatorname{Ei}{\left({\color{red}{u \ln{\left(a \right)}}} \right)}}{2 \ln{\left(2 \right)}}$$

Recorde que $$$u=4^{x}$$$:

$$\frac{\operatorname{Ei}{\left(\ln{\left(a \right)} {\color{red}{u}} \right)}}{2 \ln{\left(2 \right)}} = \frac{\operatorname{Ei}{\left(\ln{\left(a \right)} {\color{red}{4^{x}}} \right)}}{2 \ln{\left(2 \right)}}$$

Portanto,

$$\int{a^{4^{x}} d x} = \frac{\operatorname{Ei}{\left(4^{x} \ln{\left(a \right)} \right)}}{2 \ln{\left(2 \right)}}$$

Adicione a constante de integração:

$$\int{a^{4^{x}} d x} = \frac{\operatorname{Ei}{\left(4^{x} \ln{\left(a \right)} \right)}}{2 \ln{\left(2 \right)}}+C$$

$$$\int a^{4^{x}}\, dx = \frac{\operatorname{Ei}{\left(4^{x} \ln\left(a\right) \right)}}{2 \ln\left(2\right)} + C$$$A