Integral de $$$\left(\frac{3}{4}\right)^{x^{2}}$$$

Encontre $$$\int \left(\frac{3}{4}\right)^{x^{2}}\, dx$$$.

Alterar a base:

$${\color{red}{\int{\left(\frac{3}{4}\right)^{x^{2}} d x}}} = {\color{red}{\int{e^{x^{2} \ln{\left(\frac{3}{4} \right)}} d x}}}$$

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

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

A integral torna-se

$${\color{red}{\int{e^{x^{2} \ln{\left(\frac{3}{4} \right)}} d x}}} = {\color{red}{\int{\frac{e^{- u^{2}}}{\sqrt{- \ln{\left(3 \right)} + 2 \ln{\left(2 \right)}}} d u}}}$$

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

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

Esta integral (Função erro) não possui forma fechada:

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

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

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

Portanto,

$$\int{\left(\frac{3}{4}\right)^{x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \sqrt{- \ln{\left(3 \right)} + 2 \ln{\left(2 \right)}} \right)}}{2 \sqrt{- \ln{\left(3 \right)} + 2 \ln{\left(2 \right)}}}$$

Adicione a constante de integração:

$$\int{\left(\frac{3}{4}\right)^{x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \sqrt{- \ln{\left(3 \right)} + 2 \ln{\left(2 \right)}} \right)}}{2 \sqrt{- \ln{\left(3 \right)} + 2 \ln{\left(2 \right)}}}+C$$

$$$\int \left(\frac{3}{4}\right)^{x^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)} \right)}}{2 \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}} + C$$$A