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Integralen av $$$\frac{e^{- t^{2} x^{2}}}{t^{2}}$$$ med avseende på $$$x$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \frac{e^{- t^{2} x^{2}}}{t^{2}}\, dx$$$.
Lösning
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{t^{2}}$$$ och $$$f{\left(x \right)} = e^{- t^{2} x^{2}}$$$:
$${\color{red}{\int{\frac{e^{- t^{2} x^{2}}}{t^{2}} d x}}} = {\color{red}{\frac{\int{e^{- t^{2} x^{2}} d x}}{t^{2}}}}$$
Låt $$$u=x \left|{t}\right|$$$ vara.
Då $$$du=\left(x \left|{t}\right|\right)^{\prime }dx = \left|{t}\right| dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{\left|{t}\right|}$$$.
Integralen blir
$$\frac{{\color{red}{\int{e^{- t^{2} x^{2}} d x}}}}{t^{2}} = \frac{{\color{red}{\int{\frac{e^{- u^{2}}}{\left|{t}\right|} d u}}}}{t^{2}}$$
Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{\left|{t}\right|}$$$ och $$$f{\left(u \right)} = e^{- u^{2}}$$$:
$$\frac{{\color{red}{\int{\frac{e^{- u^{2}}}{\left|{t}\right|} d u}}}}{t^{2}} = \frac{{\color{red}{\frac{\int{e^{- u^{2}} d u}}{\left|{t}\right|}}}}{t^{2}}$$
Denna integral (Felintegral) har ingen sluten form:
$$\frac{{\color{red}{\int{e^{- u^{2}} d u}}}}{t^{2} \left|{t}\right|} = \frac{{\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(u \right)}}{2}\right)}}}{t^{2} \left|{t}\right|}$$
Kom ihåg att $$$u=x \left|{t}\right|$$$:
$$\frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{u}} \right)}}{2 t^{2} \left|{t}\right|} = \frac{\sqrt{\pi} \operatorname{erf}{\left({\color{red}{x \left|{t}\right|}} \right)}}{2 t^{2} \left|{t}\right|}$$
Alltså,
$$\int{\frac{e^{- t^{2} x^{2}}}{t^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{t}\right| \right)}}{2 t^{2} \left|{t}\right|}$$
Lägg till integrationskonstanten:
$$\int{\frac{e^{- t^{2} x^{2}}}{t^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{t}\right| \right)}}{2 t^{2} \left|{t}\right|}+C$$
Svar
$$$\int \frac{e^{- t^{2} x^{2}}}{t^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \left|{t}\right| \right)}}{2 t^{2} \left|{t}\right|} + C$$$A