Integralen av $$$x^{2} \sqrt{3 - x}$$$

Bestäm $$$\int x^{2} \sqrt{3 - x}\, dx$$$.

Låt $$$u=3 - x$$$ vara.

Då $$$du=\left(3 - x\right)^{\prime }dx = - dx$$$ (stegen kan ses »), och vi har att $$$dx = - du$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=-1$$$ och $$$f{\left(u \right)} = \sqrt{u} \left(u - 3\right)^{2}$$$:

$${\color{red}{\int{\left(- \sqrt{u} \left(u - 3\right)^{2}\right)d u}}} = {\color{red}{\left(- \int{\sqrt{u} \left(u - 3\right)^{2} d u}\right)}}$$

Expand the expression:

$$- {\color{red}{\int{\sqrt{u} \left(u - 3\right)^{2} d u}}} = - {\color{red}{\int{\left(u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 \sqrt{u}\right)d u}}}$$

Integrera termvis:

$$- {\color{red}{\int{\left(u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 \sqrt{u}\right)d u}}} = - {\color{red}{\left(\int{9 \sqrt{u} d u} - \int{6 u^{\frac{3}{2}} d u} + \int{u^{\frac{5}{2}} d u}\right)}}$$

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{5}{2}$$$:

$$- \int{9 \sqrt{u} d u} + \int{6 u^{\frac{3}{2}} d u} - {\color{red}{\int{u^{\frac{5}{2}} d u}}}=- \int{9 \sqrt{u} d u} + \int{6 u^{\frac{3}{2}} d u} - {\color{red}{\frac{u^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=- \int{9 \sqrt{u} d u} + \int{6 u^{\frac{3}{2}} d u} - {\color{red}{\left(\frac{2 u^{\frac{7}{2}}}{7}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=6$$$ och $$$f{\left(u \right)} = u^{\frac{3}{2}}$$$:

$$- \frac{2 u^{\frac{7}{2}}}{7} - \int{9 \sqrt{u} d u} + {\color{red}{\int{6 u^{\frac{3}{2}} d u}}} = - \frac{2 u^{\frac{7}{2}}}{7} - \int{9 \sqrt{u} d u} + {\color{red}{\left(6 \int{u^{\frac{3}{2}} d u}\right)}}$$

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{3}{2}$$$:

$$- \frac{2 u^{\frac{7}{2}}}{7} - \int{9 \sqrt{u} d u} + 6 {\color{red}{\int{u^{\frac{3}{2}} d u}}}=- \frac{2 u^{\frac{7}{2}}}{7} - \int{9 \sqrt{u} d u} + 6 {\color{red}{\frac{u^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=- \frac{2 u^{\frac{7}{2}}}{7} - \int{9 \sqrt{u} d u} + 6 {\color{red}{\left(\frac{2 u^{\frac{5}{2}}}{5}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=9$$$ och $$$f{\left(u \right)} = \sqrt{u}$$$:

$$- \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - {\color{red}{\int{9 \sqrt{u} d u}}} = - \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - {\color{red}{\left(9 \int{\sqrt{u} d u}\right)}}$$

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{1}{2}$$$:

$$- \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - 9 {\color{red}{\int{\sqrt{u} d u}}}=- \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - 9 {\color{red}{\int{u^{\frac{1}{2}} d u}}}=- \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - 9 {\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}=- \frac{2 u^{\frac{7}{2}}}{7} + \frac{12 u^{\frac{5}{2}}}{5} - 9 {\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}$$

Kom ihåg att $$$u=3 - x$$$:

$$- 6 {\color{red}{u}}^{\frac{3}{2}} + \frac{12 {\color{red}{u}}^{\frac{5}{2}}}{5} - \frac{2 {\color{red}{u}}^{\frac{7}{2}}}{7} = - 6 {\color{red}{\left(3 - x\right)}}^{\frac{3}{2}} + \frac{12 {\color{red}{\left(3 - x\right)}}^{\frac{5}{2}}}{5} - \frac{2 {\color{red}{\left(3 - x\right)}}^{\frac{7}{2}}}{7}$$

Alltså,

$$\int{x^{2} \sqrt{3 - x} d x} = - \frac{2 \left(3 - x\right)^{\frac{7}{2}}}{7} + \frac{12 \left(3 - x\right)^{\frac{5}{2}}}{5} - 6 \left(3 - x\right)^{\frac{3}{2}}$$

Förenkla:

$$\int{x^{2} \sqrt{3 - x} d x} = \frac{2 \left(3 - x\right)^{\frac{3}{2}} \left(- 42 x - 5 \left(3 - x\right)^{2} + 21\right)}{35}$$

Lägg till integrationskonstanten:

$$\int{x^{2} \sqrt{3 - x} d x} = \frac{2 \left(3 - x\right)^{\frac{3}{2}} \left(- 42 x - 5 \left(3 - x\right)^{2} + 21\right)}{35}+C$$

$$$\int x^{2} \sqrt{3 - x}\, dx = \frac{2 \left(3 - x\right)^{\frac{3}{2}} \left(- 42 x - 5 \left(3 - x\right)^{2} + 21\right)}{35} + C$$$A