Integraal van $$$2 x^{2} + 10 x + 5$$$

Bepaal $$$\int \left(2 x^{2} + 10 x + 5\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(2 x^{2} + 10 x + 5\right)d x}}} = {\color{red}{\left(\int{5 d x} + \int{10 x d x} + \int{2 x^{2} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=5$$$:

$$\int{10 x d x} + \int{2 x^{2} d x} + {\color{red}{\int{5 d x}}} = \int{10 x d x} + \int{2 x^{2} d x} + {\color{red}{\left(5 x\right)}}$$

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

$$5 x + \int{10 x d x} + {\color{red}{\int{2 x^{2} d x}}} = 5 x + \int{10 x d x} + {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$5 x + \int{10 x d x} + 2 {\color{red}{\int{x^{2} d x}}}=5 x + \int{10 x d x} + 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=5 x + \int{10 x d x} + 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=10$$$ en $$$f{\left(x \right)} = x$$$:

$$\frac{2 x^{3}}{3} + 5 x + {\color{red}{\int{10 x d x}}} = \frac{2 x^{3}}{3} + 5 x + {\color{red}{\left(10 \int{x d x}\right)}}$$

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

$$\frac{2 x^{3}}{3} + 5 x + 10 {\color{red}{\int{x d x}}}=\frac{2 x^{3}}{3} + 5 x + 10 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{2 x^{3}}{3} + 5 x + 10 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Dus,

$$\int{\left(2 x^{2} + 10 x + 5\right)d x} = \frac{2 x^{3}}{3} + 5 x^{2} + 5 x$$

Vereenvoudig:

$$\int{\left(2 x^{2} + 10 x + 5\right)d x} = \frac{x \left(2 x^{2} + 15 x + 15\right)}{3}$$

Voeg de integratieconstante toe:

$$\int{\left(2 x^{2} + 10 x + 5\right)d x} = \frac{x \left(2 x^{2} + 15 x + 15\right)}{3}+C$$

$$$\int \left(2 x^{2} + 10 x + 5\right)\, dx = \frac{x \left(2 x^{2} + 15 x + 15\right)}{3} + C$$$A