Integralen av $$$e^{x^{2}} - 1$$$

Bestäm $$$\int \left(e^{x^{2}} - 1\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(e^{x^{2}} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{e^{x^{2}} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\int{e^{x^{2}} d x} - {\color{red}{\int{1 d x}}} = \int{e^{x^{2}} d x} - {\color{red}{x}}$$

Denna integral (Imaginära felintegralen) har ingen sluten form:

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

Alltså,

$$\int{\left(e^{x^{2}} - 1\right)d x} = - x + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2}$$

Lägg till integrationskonstanten:

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

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