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

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

Expand the expression:

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

Integrera termvis:

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

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

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

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

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=10$$$ och $$$f{\left(x \right)} = x^{\frac{3}{2}}$$$:

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{3}{2}$$$:

$$- 2 x^{\frac{3}{2}} + 10 {\color{red}{\int{x^{\frac{3}{2}} d x}}}=- 2 x^{\frac{3}{2}} + 10 {\color{red}{\frac{x^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=- 2 x^{\frac{3}{2}} + 10 {\color{red}{\left(\frac{2 x^{\frac{5}{2}}}{5}\right)}}$$

Alltså,

$$\int{\sqrt{x} \left(10 x - 3\right) d x} = 4 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}}$$

Förenkla:

$$\int{\sqrt{x} \left(10 x - 3\right) d x} = x^{\frac{3}{2}} \left(4 x - 2\right)$$

Lägg till integrationskonstanten:

$$\int{\sqrt{x} \left(10 x - 3\right) d x} = x^{\frac{3}{2}} \left(4 x - 2\right)+C$$

$$$\int \sqrt{x} \left(10 x - 3\right)\, dx = x^{\frac{3}{2}} \left(4 x - 2\right) + C$$$A