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Integraal van $$$- \frac{x^{3}}{e^{6}} + x e^{2}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x}}} = {\color{red}{\left(\int{x e^{2} d x} - \int{\frac{x^{3}}{e^{6}} d x}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=e^{2}$$$ en $$$f{\left(x \right)} = x$$$:
$$- \int{\frac{x^{3}}{e^{6}} d x} + {\color{red}{\int{x e^{2} d x}}} = - \int{\frac{x^{3}}{e^{6}} d x} + {\color{red}{e^{2} \int{x d x}}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:
$$- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\int{x d x}}}=- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=e^{-6}$$$ en $$$f{\left(x \right)} = x^{3}$$$:
$$\frac{x^{2} e^{2}}{2} - {\color{red}{\int{\frac{x^{3}}{e^{6}} d x}}} = \frac{x^{2} e^{2}}{2} - {\color{red}{\frac{\int{x^{3} d x}}{e^{6}}}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=3$$$:
$$\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\int{x^{3} d x}}}}{e^{6}}=\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\frac{x^{1 + 3}}{1 + 3}}}}{e^{6}}=\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\left(\frac{x^{4}}{4}\right)}}}{e^{6}}$$
Dus,
$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = - \frac{x^{4}}{4 e^{6}} + \frac{x^{2} e^{2}}{2}$$
Vereenvoudig:
$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}}$$
Voeg de integratieconstante toe:
$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}}+C$$
Antwoord
$$$\int \left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)\, dx = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}} + C$$$A