Integralen av $$$\frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f}$$$ med avseende på $$$x$$$

Bestäm $$$\int \frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{d}{f}$$$ och $$$f{\left(x \right)} = 7 x^{3} - 13 x^{2} - 6$$$:

$${\color{red}{\int{\frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f} d x}}} = {\color{red}{\frac{d \int{\left(7 x^{3} - 13 x^{2} - 6\right)d x}}{f}}}$$

Integrera termvis:

$$\frac{d {\color{red}{\int{\left(7 x^{3} - 13 x^{2} - 6\right)d x}}}}{f} = \frac{d {\color{red}{\left(- \int{6 d x} - \int{13 x^{2} d x} + \int{7 x^{3} d x}\right)}}}{f}$$

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

$$\frac{d \left(- \int{13 x^{2} d x} + \int{7 x^{3} d x} - {\color{red}{\int{6 d x}}}\right)}{f} = \frac{d \left(- \int{13 x^{2} d x} + \int{7 x^{3} d x} - {\color{red}{\left(6 x\right)}}\right)}{f}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=13$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{d \left(- 6 x + \int{7 x^{3} d x} - {\color{red}{\int{13 x^{2} d x}}}\right)}{f} = \frac{d \left(- 6 x + \int{7 x^{3} d x} - {\color{red}{\left(13 \int{x^{2} d x}\right)}}\right)}{f}$$

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

$$\frac{d \left(- 6 x + \int{7 x^{3} d x} - 13 {\color{red}{\int{x^{2} d x}}}\right)}{f}=\frac{d \left(- 6 x + \int{7 x^{3} d x} - 13 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}\right)}{f}=\frac{d \left(- 6 x + \int{7 x^{3} d x} - 13 {\color{red}{\left(\frac{x^{3}}{3}\right)}}\right)}{f}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=7$$$ och $$$f{\left(x \right)} = x^{3}$$$:

$$\frac{d \left(- \frac{13 x^{3}}{3} - 6 x + {\color{red}{\int{7 x^{3} d x}}}\right)}{f} = \frac{d \left(- \frac{13 x^{3}}{3} - 6 x + {\color{red}{\left(7 \int{x^{3} d x}\right)}}\right)}{f}$$

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

$$\frac{d \left(- \frac{13 x^{3}}{3} - 6 x + 7 {\color{red}{\int{x^{3} d x}}}\right)}{f}=\frac{d \left(- \frac{13 x^{3}}{3} - 6 x + 7 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}\right)}{f}=\frac{d \left(- \frac{13 x^{3}}{3} - 6 x + 7 {\color{red}{\left(\frac{x^{4}}{4}\right)}}\right)}{f}$$

Alltså,

$$\int{\frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f} d x} = \frac{d \left(\frac{7 x^{4}}{4} - \frac{13 x^{3}}{3} - 6 x\right)}{f}$$

Förenkla:

$$\int{\frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f} d x} = \frac{d x \left(21 x^{3} - 52 x^{2} - 72\right)}{12 f}$$

Lägg till integrationskonstanten:

$$\int{\frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f} d x} = \frac{d x \left(21 x^{3} - 52 x^{2} - 72\right)}{12 f}+C$$

$$$\int \frac{d \left(7 x^{3} - 13 x^{2} - 6\right)}{f}\, dx = \frac{d x \left(21 x^{3} - 52 x^{2} - 72\right)}{12 f} + C$$$A