Integraal van $$$\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}$$$

Bepaal $$$\int \frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}\, dx$$$.

Vereenvoudig de integraand:

$${\color{red}{\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x}}} = {\color{red}{\int{\frac{25}{4 x^{2} - 12 x + 9} d x}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=25$$$ en $$$f{\left(x \right)} = \frac{1}{4 x^{2} - 12 x + 9}$$$:

$${\color{red}{\int{\frac{25}{4 x^{2} - 12 x + 9} d x}}} = {\color{red}{\left(25 \int{\frac{1}{4 x^{2} - 12 x + 9} d x}\right)}}$$

Voltooi het kwadraat (stappen zijn te zien »): $$$4 x^{2} - 12 x + 9 = \left(2 x - 3\right)^{2}$$$:

$$25 {\color{red}{\int{\frac{1}{4 x^{2} - 12 x + 9} d x}}} = 25 {\color{red}{\int{\frac{1}{\left(2 x - 3\right)^{2}} d x}}}$$

Zij $$$u=2 x - 3$$$.

Dan $$$du=\left(2 x - 3\right)^{\prime }dx = 2 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{2}$$$.

Dus,

$$25 {\color{red}{\int{\frac{1}{\left(2 x - 3\right)^{2}} d x}}} = 25 {\color{red}{\int{\frac{1}{2 u^{2}} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$:

$$25 {\color{red}{\int{\frac{1}{2 u^{2}} d u}}} = 25 {\color{red}{\left(\frac{\int{\frac{1}{u^{2}} d u}}{2}\right)}}$$

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

$$\frac{25 {\color{red}{\int{\frac{1}{u^{2}} d u}}}}{2}=\frac{25 {\color{red}{\int{u^{-2} d u}}}}{2}=\frac{25 {\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}}{2}=\frac{25 {\color{red}{\left(- u^{-1}\right)}}}{2}=\frac{25 {\color{red}{\left(- \frac{1}{u}\right)}}}{2}$$

We herinneren eraan dat $$$u=2 x - 3$$$:

$$- \frac{25 {\color{red}{u}}^{-1}}{2} = - \frac{25 {\color{red}{\left(2 x - 3\right)}}^{-1}}{2}$$

Dus,

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{2 \left(2 x - 3\right)}$$

Vereenvoudig:

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{4 x - 6}$$

Voeg de integratieconstante toe:

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{4 x - 6}+C$$

$$$\int \frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}\, dx = - \frac{25}{4 x - 6} + C$$$A