Integraal van $$$\frac{x}{\sqrt[4]{1 - x}}$$$

Bepaal $$$\int \frac{x}{\sqrt[4]{1 - x}}\, dx$$$.

Zij $$$u=1 - x$$$.

Dan $$$du=\left(1 - x\right)^{\prime }dx = - dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = - du$$$.

De integraal wordt

$${\color{red}{\int{\frac{x}{\sqrt[4]{1 - x}} d x}}} = {\color{red}{\int{\frac{u - 1}{\sqrt[4]{u}} d u}}}$$

Expand the expression:

$${\color{red}{\int{\frac{u - 1}{\sqrt[4]{u}} d u}}} = {\color{red}{\int{\left(u^{\frac{3}{4}} - \frac{1}{\sqrt[4]{u}}\right)d u}}}$$

Integreer termgewijs:

$${\color{red}{\int{\left(u^{\frac{3}{4}} - \frac{1}{\sqrt[4]{u}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt[4]{u}} d u} + \int{u^{\frac{3}{4}} d u}\right)}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=\frac{3}{4}$$$:

$$- \int{\frac{1}{\sqrt[4]{u}} d u} + {\color{red}{\int{u^{\frac{3}{4}} d u}}}=- \int{\frac{1}{\sqrt[4]{u}} d u} + {\color{red}{\frac{u^{\frac{3}{4} + 1}}{\frac{3}{4} + 1}}}=- \int{\frac{1}{\sqrt[4]{u}} d u} + {\color{red}{\left(\frac{4 u^{\frac{7}{4}}}{7}\right)}}$$

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

$$\frac{4 u^{\frac{7}{4}}}{7} - {\color{red}{\int{\frac{1}{\sqrt[4]{u}} d u}}}=\frac{4 u^{\frac{7}{4}}}{7} - {\color{red}{\int{u^{- \frac{1}{4}} d u}}}=\frac{4 u^{\frac{7}{4}}}{7} - {\color{red}{\frac{u^{- \frac{1}{4} + 1}}{- \frac{1}{4} + 1}}}=\frac{4 u^{\frac{7}{4}}}{7} - {\color{red}{\left(\frac{4 u^{\frac{3}{4}}}{3}\right)}}$$

We herinneren eraan dat $$$u=1 - x$$$:

$$- \frac{4 {\color{red}{u}}^{\frac{3}{4}}}{3} + \frac{4 {\color{red}{u}}^{\frac{7}{4}}}{7} = - \frac{4 {\color{red}{\left(1 - x\right)}}^{\frac{3}{4}}}{3} + \frac{4 {\color{red}{\left(1 - x\right)}}^{\frac{7}{4}}}{7}$$

Dus,

$$\int{\frac{x}{\sqrt[4]{1 - x}} d x} = \frac{4 \left(1 - x\right)^{\frac{7}{4}}}{7} - \frac{4 \left(1 - x\right)^{\frac{3}{4}}}{3}$$

Vereenvoudig:

$$\int{\frac{x}{\sqrt[4]{1 - x}} d x} = \frac{4 \left(1 - x\right)^{\frac{3}{4}} \left(- 3 x - 4\right)}{21}$$

Voeg de integratieconstante toe:

$$\int{\frac{x}{\sqrt[4]{1 - x}} d x} = \frac{4 \left(1 - x\right)^{\frac{3}{4}} \left(- 3 x - 4\right)}{21}+C$$

$$$\int \frac{x}{\sqrt[4]{1 - x}}\, dx = \frac{4 \left(1 - x\right)^{\frac{3}{4}} \left(- 3 x - 4\right)}{21} + C$$$A