Integralen av $$$x^{3} \left(3 - x\right)^{3}$$$

Bestäm $$$\int x^{3} \left(3 - x\right)^{3}\, dx$$$.

Expand the expression:

$${\color{red}{\int{x^{3} \left(3 - x\right)^{3} d x}}} = {\color{red}{\int{\left(- x^{6} + 9 x^{5} - 27 x^{4} + 27 x^{3}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(- x^{6} + 9 x^{5} - 27 x^{4} + 27 x^{3}\right)d x}}} = {\color{red}{\left(\int{27 x^{3} d x} - \int{27 x^{4} d x} + \int{9 x^{5} d x} - \int{x^{6} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=6$$$:

$$\int{27 x^{3} d x} - \int{27 x^{4} d x} + \int{9 x^{5} d x} - {\color{red}{\int{x^{6} d x}}}=\int{27 x^{3} d x} - \int{27 x^{4} d x} + \int{9 x^{5} d x} - {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=\int{27 x^{3} d x} - \int{27 x^{4} d x} + \int{9 x^{5} d x} - {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=27$$$ och $$$f{\left(x \right)} = x^{4}$$$:

$$- \frac{x^{7}}{7} + \int{27 x^{3} d x} + \int{9 x^{5} d x} - {\color{red}{\int{27 x^{4} d x}}} = - \frac{x^{7}}{7} + \int{27 x^{3} d x} + \int{9 x^{5} d x} - {\color{red}{\left(27 \int{x^{4} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=4$$$:

$$- \frac{x^{7}}{7} + \int{27 x^{3} d x} + \int{9 x^{5} d x} - 27 {\color{red}{\int{x^{4} d x}}}=- \frac{x^{7}}{7} + \int{27 x^{3} d x} + \int{9 x^{5} d x} - 27 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \frac{x^{7}}{7} + \int{27 x^{3} d x} + \int{9 x^{5} d x} - 27 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=9$$$ och $$$f{\left(x \right)} = x^{5}$$$:

$$- \frac{x^{7}}{7} - \frac{27 x^{5}}{5} + \int{27 x^{3} d x} + {\color{red}{\int{9 x^{5} d x}}} = - \frac{x^{7}}{7} - \frac{27 x^{5}}{5} + \int{27 x^{3} d x} + {\color{red}{\left(9 \int{x^{5} d x}\right)}}$$

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

$$- \frac{x^{7}}{7} - \frac{27 x^{5}}{5} + \int{27 x^{3} d x} + 9 {\color{red}{\int{x^{5} d x}}}=- \frac{x^{7}}{7} - \frac{27 x^{5}}{5} + \int{27 x^{3} d x} + 9 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=- \frac{x^{7}}{7} - \frac{27 x^{5}}{5} + \int{27 x^{3} d x} + 9 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=27$$$ och $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + {\color{red}{\int{27 x^{3} d x}}} = - \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + {\color{red}{\left(27 \int{x^{3} d x}\right)}}$$

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

$$- \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + 27 {\color{red}{\int{x^{3} d x}}}=- \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + 27 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + 27 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Alltså,

$$\int{x^{3} \left(3 - x\right)^{3} d x} = - \frac{x^{7}}{7} + \frac{3 x^{6}}{2} - \frac{27 x^{5}}{5} + \frac{27 x^{4}}{4}$$

Förenkla:

$$\int{x^{3} \left(3 - x\right)^{3} d x} = \frac{x^{4} \left(- 20 x^{3} + 210 x^{2} - 756 x + 945\right)}{140}$$

Lägg till integrationskonstanten:

$$\int{x^{3} \left(3 - x\right)^{3} d x} = \frac{x^{4} \left(- 20 x^{3} + 210 x^{2} - 756 x + 945\right)}{140}+C$$

$$$\int x^{3} \left(3 - x\right)^{3}\, dx = \frac{x^{4} \left(- 20 x^{3} + 210 x^{2} - 756 x + 945\right)}{140} + C$$$A