Integralen av $$$e^{t^{2}} - e^{- t^{2}}$$$

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

Integrera termvis:

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

Denna integral (Felintegral) har ingen sluten form:

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

Denna integral (Imaginära felintegralen) har ingen sluten form:

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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