Integralen av $$$\frac{7 x}{12} - 6$$$

Bestäm $$$\int \left(\frac{7 x}{12} - 6\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(\frac{7 x}{12} - 6\right)d x}}} = {\color{red}{\left(- \int{6 d x} + \int{\frac{7 x}{12} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=6$$$:

$$\int{\frac{7 x}{12} d x} - {\color{red}{\int{6 d x}}} = \int{\frac{7 x}{12} d x} - {\color{red}{\left(6 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{7}{12}$$$ och $$$f{\left(x \right)} = x$$$:

$$- 6 x + {\color{red}{\int{\frac{7 x}{12} d x}}} = - 6 x + {\color{red}{\left(\frac{7 \int{x d x}}{12}\right)}}$$

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

$$- 6 x + \frac{7 {\color{red}{\int{x d x}}}}{12}=- 6 x + \frac{7 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{12}=- 6 x + \frac{7 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{12}$$

Alltså,

$$\int{\left(\frac{7 x}{12} - 6\right)d x} = \frac{7 x^{2}}{24} - 6 x$$

Förenkla:

$$\int{\left(\frac{7 x}{12} - 6\right)d x} = \frac{x \left(7 x - 144\right)}{24}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{7 x}{12} - 6\right)d x} = \frac{x \left(7 x - 144\right)}{24}+C$$

$$$\int \left(\frac{7 x}{12} - 6\right)\, dx = \frac{x \left(7 x - 144\right)}{24} + C$$$A