Integraal van $$$2 x^{3} \left(3 x - 2\right)$$$

Bepaal $$$\int 2 x^{3} \left(3 x - 2\right)\, dx$$$.

De invoer is herschreven: $$$\int{2 x^{3} \left(3 x - 2\right) d x}=\int{x^{3} \left(6 x - 4\right) d x}$$$.

Vereenvoudig de integraand:

$${\color{red}{\int{x^{3} \left(6 x - 4\right) d x}}} = {\color{red}{\int{2 x^{3} \left(3 x - 2\right) d x}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x^{3} \left(3 x - 2\right)$$$:

$${\color{red}{\int{2 x^{3} \left(3 x - 2\right) d x}}} = {\color{red}{\left(2 \int{x^{3} \left(3 x - 2\right) d x}\right)}}$$

Expand the expression:

$$2 {\color{red}{\int{x^{3} \left(3 x - 2\right) d x}}} = 2 {\color{red}{\int{\left(3 x^{4} - 2 x^{3}\right)d x}}}$$

Integreer termgewijs:

$$2 {\color{red}{\int{\left(3 x^{4} - 2 x^{3}\right)d x}}} = 2 {\color{red}{\left(- \int{2 x^{3} d x} + \int{3 x^{4} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x^{3}$$$:

$$2 \int{3 x^{4} d x} - 2 {\color{red}{\int{2 x^{3} d x}}} = 2 \int{3 x^{4} d x} - 2 {\color{red}{\left(2 \int{x^{3} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=3$$$:

$$2 \int{3 x^{4} d x} - 4 {\color{red}{\int{x^{3} d x}}}=2 \int{3 x^{4} d x} - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=2 \int{3 x^{4} d x} - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=3$$$ en $$$f{\left(x \right)} = x^{4}$$$:

$$- x^{4} + 2 {\color{red}{\int{3 x^{4} d x}}} = - x^{4} + 2 {\color{red}{\left(3 \int{x^{4} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=4$$$:

$$- x^{4} + 6 {\color{red}{\int{x^{4} d x}}}=- x^{4} + 6 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- x^{4} + 6 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Dus,

$$\int{x^{3} \left(6 x - 4\right) d x} = \frac{6 x^{5}}{5} - x^{4}$$

Vereenvoudig:

$$\int{x^{3} \left(6 x - 4\right) d x} = \frac{x^{4} \left(6 x - 5\right)}{5}$$

Voeg de integratieconstante toe:

$$\int{x^{3} \left(6 x - 4\right) d x} = \frac{x^{4} \left(6 x - 5\right)}{5}+C$$

$$$\int 2 x^{3} \left(3 x - 2\right)\, dx = \frac{x^{4} \left(6 x - 5\right)}{5} + C$$$A