Integralen av $$$x^{2} \left(x^{3} - 5\right)^{7}$$$

Bestäm $$$\int x^{2} \left(x^{3} - 5\right)^{7}\, dx$$$.

Låt $$$u=x^{3} - 5$$$ vara.

Då $$$du=\left(x^{3} - 5\right)^{\prime }dx = 3 x^{2} dx$$$ (stegen kan ses »), och vi har att $$$x^{2} dx = \frac{du}{3}$$$.

Integralen blir

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(u \right)} = u^{7}$$$:

$${\color{red}{\int{\frac{u^{7}}{3} d u}}} = {\color{red}{\left(\frac{\int{u^{7} d u}}{3}\right)}}$$

Tillämpa potensregeln $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=7$$$:

$$\frac{{\color{red}{\int{u^{7} d u}}}}{3}=\frac{{\color{red}{\frac{u^{1 + 7}}{1 + 7}}}}{3}=\frac{{\color{red}{\left(\frac{u^{8}}{8}\right)}}}{3}$$

Kom ihåg att $$$u=x^{3} - 5$$$:

$$\frac{{\color{red}{u}}^{8}}{24} = \frac{{\color{red}{\left(x^{3} - 5\right)}}^{8}}{24}$$

Alltså,

$$\int{x^{2} \left(x^{3} - 5\right)^{7} d x} = \frac{\left(x^{3} - 5\right)^{8}}{24}$$

Lägg till integrationskonstanten:

$$\int{x^{2} \left(x^{3} - 5\right)^{7} d x} = \frac{\left(x^{3} - 5\right)^{8}}{24}+C$$

$$$\int x^{2} \left(x^{3} - 5\right)^{7}\, dx = \frac{\left(x^{3} - 5\right)^{8}}{24} + C$$$A