Integral de $$$e^{t^{2}} - e^{- t^{2}}$$$

Encontre $$$\int \left(e^{t^{2}} - e^{- t^{2}}\right)\, dt$$$.

Integre termo a termo:

$${\color{red}{\int{\left(e^{t^{2}} - e^{- t^{2}}\right)d t}}} = {\color{red}{\left(- \int{e^{- t^{2}} d t} + \int{e^{t^{2}} d t}\right)}}$$

Esta integral (Função erro) não possui forma fechada:

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

Esta integral (Função erro imaginária) não possui forma fechada:

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

Portanto,

$$\int{\left(e^{t^{2}} - e^{- t^{2}}\right)d t} = - \frac{\sqrt{\pi} \operatorname{erf}{\left(t \right)}}{2} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(t \right)}}{2}$$

Simplifique:

$$\int{\left(e^{t^{2}} - e^{- t^{2}}\right)d t} = \frac{\sqrt{\pi} \left(- \operatorname{erf}{\left(t \right)} + \operatorname{erfi}{\left(t \right)}\right)}{2}$$

Adicione a constante de integração:

$$\int{\left(e^{t^{2}} - e^{- t^{2}}\right)d t} = \frac{\sqrt{\pi} \left(- \operatorname{erf}{\left(t \right)} + \operatorname{erfi}{\left(t \right)}\right)}{2}+C$$

$$$\int \left(e^{t^{2}} - e^{- t^{2}}\right)\, dt = \frac{\sqrt{\pi} \left(- \operatorname{erf}{\left(t \right)} + \operatorname{erfi}{\left(t \right)}\right)}{2} + C$$$A