Integral de $$$3^{\sqrt{2} \sqrt{x}}$$$

Encontre $$$\int 3^{\sqrt{2} \sqrt{x}}\, dx$$$.

Alterar a base:

$${\color{red}{\int{3^{\sqrt{2} \sqrt{x}} d x}}} = {\color{red}{\int{e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}} d x}}}$$

Seja $$$u=\sqrt{2} \sqrt{x} \ln{\left(3 \right)}$$$.

Então $$$du=\left(\sqrt{2} \sqrt{x} \ln{\left(3 \right)}\right)^{\prime }dx = \frac{\sqrt{2} \ln{\left(3 \right)}}{2 \sqrt{x}} dx$$$ (veja os passos »), e obtemos $$$\frac{dx}{\sqrt{x}} = \frac{\sqrt{2} du}{\ln{\left(3 \right)}}$$$.

A integral torna-se

$${\color{red}{\int{e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}} d x}}} = {\color{red}{\int{\frac{u e^{u}}{\ln{\left(3 \right)}^{2}} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{\ln{\left(3 \right)}^{2}}$$$ e $$$f{\left(u \right)} = u e^{u}$$$:

$${\color{red}{\int{\frac{u e^{u}}{\ln{\left(3 \right)}^{2}} d u}}} = {\color{red}{\frac{\int{u e^{u} d u}}{\ln{\left(3 \right)}^{2}}}}$$

Para a integral $$$\int{u e^{u} d u}$$$, use integração por partes $$$\int \operatorname{g} \operatorname{dv} = \operatorname{g}\operatorname{v} - \int \operatorname{v} \operatorname{dg}$$$.

Sejam $$$\operatorname{g}=u$$$ e $$$\operatorname{dv}=e^{u} du$$$.

Então $$$\operatorname{dg}=\left(u\right)^{\prime }du=1 du$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{u} d u}=e^{u}$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

$$\frac{{\color{red}{\int{u e^{u} d u}}}}{\ln{\left(3 \right)}^{2}}=\frac{{\color{red}{\left(u \cdot e^{u}-\int{e^{u} \cdot 1 d u}\right)}}}{\ln{\left(3 \right)}^{2}}=\frac{{\color{red}{\left(u e^{u} - \int{e^{u} d u}\right)}}}{\ln{\left(3 \right)}^{2}}$$

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

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

Recorde que $$$u=\sqrt{2} \sqrt{x} \ln{\left(3 \right)}$$$:

$$\frac{- e^{{\color{red}{u}}} + {\color{red}{u}} e^{{\color{red}{u}}}}{\ln{\left(3 \right)}^{2}} = \frac{- e^{{\color{red}{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}}} + {\color{red}{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}} e^{{\color{red}{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}}}}{\ln{\left(3 \right)}^{2}}$$

Portanto,

$$\int{3^{\sqrt{2} \sqrt{x}} d x} = \frac{\sqrt{2} \sqrt{x} e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}} \ln{\left(3 \right)} - e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}}{\ln{\left(3 \right)}^{2}}$$

Simplifique:

$$\int{3^{\sqrt{2} \sqrt{x}} d x} = \frac{\left(\sqrt{2} \sqrt{x} \ln{\left(3 \right)} - 1\right) e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}}{\ln{\left(3 \right)}^{2}}$$

Adicione a constante de integração:

$$\int{3^{\sqrt{2} \sqrt{x}} d x} = \frac{\left(\sqrt{2} \sqrt{x} \ln{\left(3 \right)} - 1\right) e^{\sqrt{2} \sqrt{x} \ln{\left(3 \right)}}}{\ln{\left(3 \right)}^{2}}+C$$

$$$\int 3^{\sqrt{2} \sqrt{x}}\, dx = \frac{\left(\sqrt{2} \sqrt{x} \ln\left(3\right) - 1\right) e^{\sqrt{2} \sqrt{x} \ln\left(3\right)}}{\ln^{2}\left(3\right)} + C$$$A