Integraal van $$$4 t^{2} - 11$$$

Bepaal $$$\int \left(4 t^{2} - 11\right)\, dt$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(4 t^{2} - 11\right)d t}}} = {\color{red}{\left(- \int{11 d t} + \int{4 t^{2} d t}\right)}}$$

Pas de constantenregel $$$\int c\, dt = c t$$$ toe met $$$c=11$$$:

$$\int{4 t^{2} d t} - {\color{red}{\int{11 d t}}} = \int{4 t^{2} d t} - {\color{red}{\left(11 t\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ toe met $$$c=4$$$ en $$$f{\left(t \right)} = t^{2}$$$:

$$- 11 t + {\color{red}{\int{4 t^{2} d t}}} = - 11 t + {\color{red}{\left(4 \int{t^{2} d t}\right)}}$$

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

$$- 11 t + 4 {\color{red}{\int{t^{2} d t}}}=- 11 t + 4 {\color{red}{\frac{t^{1 + 2}}{1 + 2}}}=- 11 t + 4 {\color{red}{\left(\frac{t^{3}}{3}\right)}}$$

Dus,

$$\int{\left(4 t^{2} - 11\right)d t} = \frac{4 t^{3}}{3} - 11 t$$

Vereenvoudig:

$$\int{\left(4 t^{2} - 11\right)d t} = \frac{t \left(4 t^{2} - 33\right)}{3}$$

Voeg de integratieconstante toe:

$$\int{\left(4 t^{2} - 11\right)d t} = \frac{t \left(4 t^{2} - 33\right)}{3}+C$$

$$$\int \left(4 t^{2} - 11\right)\, dt = \frac{t \left(4 t^{2} - 33\right)}{3} + C$$$A