Integraal van $$$\frac{23319 p}{100} - t \sqrt{a - t}$$$ met betrekking tot $$$t$$$

Bepaal $$$\int \left(\frac{23319 p}{100} - t \sqrt{a - t}\right)\, dt$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(\frac{23319 p}{100} - t \sqrt{a - t}\right)d t}}} = {\color{red}{\left(\int{\frac{23319 p}{100} d t} - \int{t \sqrt{a - t} d t}\right)}}$$

Pas de constantenregel $$$\int c\, dt = c t$$$ toe met $$$c=\frac{23319 p}{100}$$$:

$$- \int{t \sqrt{a - t} d t} + {\color{red}{\int{\frac{23319 p}{100} d t}}} = - \int{t \sqrt{a - t} d t} + {\color{red}{\left(\frac{23319 p t}{100}\right)}}$$

Zij $$$u=a - t$$$.

Dan $$$du=\left(a - t\right)^{\prime }dt = - dt$$$ (de stappen zijn te zien »), en dan geldt dat $$$dt = - du$$$.

De integraal wordt

$$\frac{23319 p t}{100} - {\color{red}{\int{t \sqrt{a - t} d t}}} = \frac{23319 p t}{100} - {\color{red}{\int{\sqrt{u} \left(- a + u\right) d u}}}$$

Expand the expression:

$$\frac{23319 p t}{100} - {\color{red}{\int{\sqrt{u} \left(- a + u\right) d u}}} = \frac{23319 p t}{100} - {\color{red}{\int{\left(- a \sqrt{u} + u^{\frac{3}{2}}\right)d u}}}$$

Integreer termgewijs:

$$\frac{23319 p t}{100} - {\color{red}{\int{\left(- a \sqrt{u} + u^{\frac{3}{2}}\right)d u}}} = \frac{23319 p t}{100} - {\color{red}{\left(\int{u^{\frac{3}{2}} d u} - \int{a \sqrt{u} d u}\right)}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=\frac{3}{2}$$$:

$$\frac{23319 p t}{100} + \int{a \sqrt{u} d u} - {\color{red}{\int{u^{\frac{3}{2}} d u}}}=\frac{23319 p t}{100} + \int{a \sqrt{u} d u} - {\color{red}{\frac{u^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=\frac{23319 p t}{100} + \int{a \sqrt{u} d u} - {\color{red}{\left(\frac{2 u^{\frac{5}{2}}}{5}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=a$$$ en $$$f{\left(u \right)} = \sqrt{u}$$$:

$$\frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5} + {\color{red}{\int{a \sqrt{u} d u}}} = \frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5} + {\color{red}{a \int{\sqrt{u} d u}}}$$

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

$$a {\color{red}{\int{\sqrt{u} d u}}} + \frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5}=a {\color{red}{\int{u^{\frac{1}{2}} d u}}} + \frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5}=a {\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}} + \frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5}=a {\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}} + \frac{23319 p t}{100} - \frac{2 u^{\frac{5}{2}}}{5}$$

We herinneren eraan dat $$$u=a - t$$$:

$$\frac{2 a {\color{red}{u}}^{\frac{3}{2}}}{3} + \frac{23319 p t}{100} - \frac{2 {\color{red}{u}}^{\frac{5}{2}}}{5} = \frac{2 a {\color{red}{\left(a - t\right)}}^{\frac{3}{2}}}{3} + \frac{23319 p t}{100} - \frac{2 {\color{red}{\left(a - t\right)}}^{\frac{5}{2}}}{5}$$

Dus,

$$\int{\left(\frac{23319 p}{100} - t \sqrt{a - t}\right)d t} = \frac{2 a \left(a - t\right)^{\frac{3}{2}}}{3} + \frac{23319 p t}{100} - \frac{2 \left(a - t\right)^{\frac{5}{2}}}{5}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{23319 p}{100} - t \sqrt{a - t}\right)d t} = \frac{2 a \left(a - t\right)^{\frac{3}{2}}}{3} + \frac{23319 p t}{100} - \frac{2 \left(a - t\right)^{\frac{5}{2}}}{5}+C$$

$$$\int \left(\frac{23319 p}{100} - t \sqrt{a - t}\right)\, dt = \left(\frac{2 a \left(a - t\right)^{\frac{3}{2}}}{3} + \frac{23319 p t}{100} - \frac{2 \left(a - t\right)^{\frac{5}{2}}}{5}\right) + C$$$A