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Integralen av $$$2 \cos{\left(x^{2} \right)}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int 2 \cos{\left(x^{2} \right)}\, dx$$$.
Lösning
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:
$${\color{red}{\int{2 \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\left(2 \int{\cos{\left(x^{2} \right)} d x}\right)}}$$
Denna integral (Fresnels cosinusintegral) har ingen sluten form:
$$2 {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} = 2 {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$
Alltså,
$$\int{2 \cos{\left(x^{2} \right)} d x} = \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)$$
Lägg till integrationskonstanten:
$$\int{2 \cos{\left(x^{2} \right)} d x} = \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)+C$$
Svar
$$$\int 2 \cos{\left(x^{2} \right)}\, dx = \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + C$$$A