Integralen av $$$3 x \left(2 - 5 x\right)$$$

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = x \left(2 - 5 x\right)$$$:

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

Expand the expression:

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

Integrera termvis:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = x^{2}$$$:

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

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

$$3 \int{2 x d x} - 15 {\color{red}{\int{x^{2} d x}}}=3 \int{2 x d x} - 15 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=3 \int{2 x d x} - 15 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

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

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

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

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

Alltså,

$$\int{3 x \left(2 - 5 x\right) d x} = - 5 x^{3} + 3 x^{2}$$

Förenkla:

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

Lägg till integrationskonstanten:

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

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