Integralen av $$$3 x^{2} - 3 - \frac{1}{2 \sqrt{x}}$$$

Bestäm $$$\int \left(3 x^{2} - 3 - \frac{1}{2 \sqrt{x}}\right)\, dx$$$.

Integrera termvis:

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

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

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

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

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

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

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

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

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

$$$\int \left(3 x^{2} - 3 - \frac{1}{2 \sqrt{x}}\right)\, dx = \left(- \sqrt{x} + x^{3} - 3 x\right) + C$$$A