Integralen av $$$i a f l n t^{3} x^{3} y \left(x^{2} + 2\right)$$$ med avseende på $$$x$$$

Bestäm $$$\int i a f l n t^{3} x^{3} y \left(x^{2} + 2\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=i a f l n t^{3} y$$$ och $$$f{\left(x \right)} = x^{3} \left(x^{2} + 2\right)$$$:

$${\color{red}{\int{i a f l n t^{3} x^{3} y \left(x^{2} + 2\right) d x}}} = {\color{red}{i a f l n t^{3} y \int{x^{3} \left(x^{2} + 2\right) d x}}}$$

Expand the expression:

$$i a f l n t^{3} y {\color{red}{\int{x^{3} \left(x^{2} + 2\right) d x}}} = i a f l n t^{3} y {\color{red}{\int{\left(x^{5} + 2 x^{3}\right)d x}}}$$

Integrera termvis:

$$i a f l n t^{3} y {\color{red}{\int{\left(x^{5} + 2 x^{3}\right)d x}}} = i a f l n t^{3} y {\color{red}{\left(\int{2 x^{3} d x} + \int{x^{5} d x}\right)}}$$

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

$$i a f l n t^{3} y \left(\int{2 x^{3} d x} + {\color{red}{\int{x^{5} d x}}}\right)=i a f l n t^{3} y \left(\int{2 x^{3} d x} + {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}\right)=i a f l n t^{3} y \left(\int{2 x^{3} d x} + {\color{red}{\left(\frac{x^{6}}{6}\right)}}\right)$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = x^{3}$$$:

$$i a f l n t^{3} y \left(\frac{x^{6}}{6} + {\color{red}{\int{2 x^{3} d x}}}\right) = i a f l n t^{3} y \left(\frac{x^{6}}{6} + {\color{red}{\left(2 \int{x^{3} d x}\right)}}\right)$$

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

$$i a f l n t^{3} y \left(\frac{x^{6}}{6} + 2 {\color{red}{\int{x^{3} d x}}}\right)=i a f l n t^{3} y \left(\frac{x^{6}}{6} + 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}\right)=i a f l n t^{3} y \left(\frac{x^{6}}{6} + 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}\right)$$

Alltså,

$$\int{i a f l n t^{3} x^{3} y \left(x^{2} + 2\right) d x} = i a f l n t^{3} y \left(\frac{x^{6}}{6} + \frac{x^{4}}{2}\right)$$

Förenkla:

$$\int{i a f l n t^{3} x^{3} y \left(x^{2} + 2\right) d x} = \frac{i a f l n t^{3} x^{4} y \left(x^{2} + 3\right)}{6}$$

Lägg till integrationskonstanten:

$$\int{i a f l n t^{3} x^{3} y \left(x^{2} + 2\right) d x} = \frac{i a f l n t^{3} x^{4} y \left(x^{2} + 3\right)}{6}+C$$

$$$\int i a f l n t^{3} x^{3} y \left(x^{2} + 2\right)\, dx = \frac{i a f l n t^{3} x^{4} y \left(x^{2} + 3\right)}{6} + C$$$A