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Integralen av $$$\frac{\theta e^{2}}{2}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \frac{\theta e^{2}}{2}\, d\theta$$$.
Lösning
Tillämpa konstantfaktorregeln $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ med $$$c=\frac{e^{2}}{2}$$$ och $$$f{\left(\theta \right)} = \theta$$$:
$${\color{red}{\int{\frac{\theta e^{2}}{2} d \theta}}} = {\color{red}{\left(\frac{e^{2} \int{\theta d \theta}}{2}\right)}}$$
Tillämpa potensregeln $$$\int \theta^{n}\, d\theta = \frac{\theta^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:
$$\frac{e^{2} {\color{red}{\int{\theta d \theta}}}}{2}=\frac{e^{2} {\color{red}{\frac{\theta^{1 + 1}}{1 + 1}}}}{2}=\frac{e^{2} {\color{red}{\left(\frac{\theta^{2}}{2}\right)}}}{2}$$
Alltså,
$$\int{\frac{\theta e^{2}}{2} d \theta} = \frac{\theta^{2} e^{2}}{4}$$
Lägg till integrationskonstanten:
$$\int{\frac{\theta e^{2}}{2} d \theta} = \frac{\theta^{2} e^{2}}{4}+C$$
Svar
$$$\int \frac{\theta e^{2}}{2}\, d\theta = \frac{\theta^{2} e^{2}}{4} + C$$$A