Integralen av $$$- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}$$$

Bestäm $$$\int \left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{x^{2}} d x} - \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x}\right)}}$$

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

$$- \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x} + {\color{red}{\int{\frac{1}{x^{2}} d x}}}=- \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x} + {\color{red}{\int{x^{-2} d x}}}=- \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x} + {\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}}=- \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x} + {\color{red}{\left(- x^{-1}\right)}}=- \int{\frac{2}{\sqrt{x}} d x} - \int{3 x^{4} d x} + {\color{red}{\left(- \frac{1}{x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = x^{4}$$$:

$$- \int{\frac{2}{\sqrt{x}} d x} - {\color{red}{\int{3 x^{4} d x}}} - \frac{1}{x} = - \int{\frac{2}{\sqrt{x}} d x} - {\color{red}{\left(3 \int{x^{4} d x}\right)}} - \frac{1}{x}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=4$$$:

$$- \int{\frac{2}{\sqrt{x}} d x} - 3 {\color{red}{\int{x^{4} d x}}} - \frac{1}{x}=- \int{\frac{2}{\sqrt{x}} d x} - 3 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}} - \frac{1}{x}=- \int{\frac{2}{\sqrt{x}} d x} - 3 {\color{red}{\left(\frac{x^{5}}{5}\right)}} - \frac{1}{x}$$

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

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

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

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

Alltså,

$$\int{\left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)d x} = - 4 \sqrt{x} - \frac{3 x^{5}}{5} - \frac{1}{x}$$

Förenkla:

$$\int{\left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)d x} = \frac{- 20 x^{\frac{3}{2}} - 3 x^{6} - 5}{5 x}$$

Lägg till integrationskonstanten:

$$\int{\left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)d x} = \frac{- 20 x^{\frac{3}{2}} - 3 x^{6} - 5}{5 x}+C$$

$$$\int \left(- 3 x^{4} + \frac{1}{x^{2}} - \frac{2}{\sqrt{x}}\right)\, dx = \frac{- 20 x^{\frac{3}{2}} - 3 x^{6} - 5}{5 x} + C$$$A