Integralen av $$$x \left(20 x - 10\right) + \sqrt{3}$$$

Bestäm $$$\int \left(x \left(20 x - 10\right) + \sqrt{3}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x}}} = {\color{red}{\left(\int{\sqrt{3} d x} + \int{x \left(20 x - 10\right) d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=\sqrt{3}$$$:

$$\int{x \left(20 x - 10\right) d x} + {\color{red}{\int{\sqrt{3} d x}}} = \int{x \left(20 x - 10\right) d x} + {\color{red}{\sqrt{3} x}}$$

Förenkla integranden:

$$\sqrt{3} x + {\color{red}{\int{x \left(20 x - 10\right) d x}}} = \sqrt{3} x + {\color{red}{\int{10 x \left(2 x - 1\right) d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=10$$$ och $$$f{\left(x \right)} = x \left(2 x - 1\right)$$$:

$$\sqrt{3} x + {\color{red}{\int{10 x \left(2 x - 1\right) d x}}} = \sqrt{3} x + {\color{red}{\left(10 \int{x \left(2 x - 1\right) d x}\right)}}$$

Expand the expression:

$$\sqrt{3} x + 10 {\color{red}{\int{x \left(2 x - 1\right) d x}}} = \sqrt{3} x + 10 {\color{red}{\int{\left(2 x^{2} - x\right)d x}}}$$

Integrera termvis:

$$\sqrt{3} x + 10 {\color{red}{\int{\left(2 x^{2} - x\right)d x}}} = \sqrt{3} x + 10 {\color{red}{\left(- \int{x d x} + \int{2 x^{2} d x}\right)}}$$

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

$$\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\int{x d x}}}=\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$- 5 x^{2} + \sqrt{3} x + 10 {\color{red}{\int{2 x^{2} d x}}} = - 5 x^{2} + \sqrt{3} x + 10 {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

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

$$- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\int{x^{2} d x}}}=- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Alltså,

$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{20 x^{3}}{3} - 5 x^{2} + \sqrt{3} x$$

Förenkla:

$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3}+C$$

$$$\int \left(x \left(20 x - 10\right) + \sqrt{3}\right)\, dx = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3} + C$$$A