|
|
|
|
|
|
|
|
|
|
|
|
Integralen av $$$108 \cos{\left(x^{2} \right)}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int 108 \cos{\left(x^{2} \right)}\, dx$$$.
Lösning
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=108$$$ och $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:
$${\color{red}{\int{108 \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\left(108 \int{\cos{\left(x^{2} \right)} d x}\right)}}$$
Denna integral (Fresnels cosinusintegral) har ingen sluten form:
$$108 {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} = 108 {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$
Alltså,
$$\int{108 \cos{\left(x^{2} \right)} d x} = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)$$
Lägg till integrationskonstanten:
$$\int{108 \cos{\left(x^{2} \right)} d x} = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)+C$$
Svar
$$$\int 108 \cos{\left(x^{2} \right)}\, dx = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + C$$$A