Integralen av $$$\frac{x - 7}{823543 x^{7}}$$$

Bestäm $$$\int \frac{x - 7}{823543 x^{7}}\, dx$$$.

Inmatningen skrivs om: $$$\int{\frac{x - 7}{823543 x^{7}} d x}=\int{\frac{\frac{x}{823543} - \frac{1}{117649}}{x^{7}} d x}$$$.

Förenkla integranden:

$${\color{red}{\int{\frac{\frac{x}{823543} - \frac{1}{117649}}{x^{7}} d x}}} = {\color{red}{\int{\frac{x - 7}{823543 x^{7}} d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{823543}$$$ och $$$f{\left(x \right)} = \frac{x - 7}{x^{7}}$$$:

$${\color{red}{\int{\frac{x - 7}{823543 x^{7}} d x}}} = {\color{red}{\left(\frac{\int{\frac{x - 7}{x^{7}} d x}}{823543}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\frac{x - 7}{x^{7}} d x}}}}{823543} = \frac{{\color{red}{\int{\left(\frac{1}{x^{6}} - \frac{7}{x^{7}}\right)d x}}}}{823543}$$

Integrera termvis:

$$\frac{{\color{red}{\int{\left(\frac{1}{x^{6}} - \frac{7}{x^{7}}\right)d x}}}}{823543} = \frac{{\color{red}{\left(- \int{\frac{7}{x^{7}} d x} + \int{\frac{1}{x^{6}} d x}\right)}}}{823543}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-6$$$:

$$- \frac{\int{\frac{7}{x^{7}} d x}}{823543} + \frac{{\color{red}{\int{\frac{1}{x^{6}} d x}}}}{823543}=- \frac{\int{\frac{7}{x^{7}} d x}}{823543} + \frac{{\color{red}{\int{x^{-6} d x}}}}{823543}=- \frac{\int{\frac{7}{x^{7}} d x}}{823543} + \frac{{\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}}}{823543}=- \frac{\int{\frac{7}{x^{7}} d x}}{823543} + \frac{{\color{red}{\left(- \frac{x^{-5}}{5}\right)}}}{823543}=- \frac{\int{\frac{7}{x^{7}} d x}}{823543} + \frac{{\color{red}{\left(- \frac{1}{5 x^{5}}\right)}}}{823543}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=7$$$ och $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$:

$$- \frac{{\color{red}{\int{\frac{7}{x^{7}} d x}}}}{823543} - \frac{1}{4117715 x^{5}} = - \frac{{\color{red}{\left(7 \int{\frac{1}{x^{7}} d x}\right)}}}{823543} - \frac{1}{4117715 x^{5}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-7$$$:

$$- \frac{{\color{red}{\int{\frac{1}{x^{7}} d x}}}}{117649} - \frac{1}{4117715 x^{5}}=- \frac{{\color{red}{\int{x^{-7} d x}}}}{117649} - \frac{1}{4117715 x^{5}}=- \frac{{\color{red}{\frac{x^{-7 + 1}}{-7 + 1}}}}{117649} - \frac{1}{4117715 x^{5}}=- \frac{{\color{red}{\left(- \frac{x^{-6}}{6}\right)}}}{117649} - \frac{1}{4117715 x^{5}}=- \frac{{\color{red}{\left(- \frac{1}{6 x^{6}}\right)}}}{117649} - \frac{1}{4117715 x^{5}}$$

Alltså,

$$\int{\frac{\frac{x}{823543} - \frac{1}{117649}}{x^{7}} d x} = - \frac{1}{4117715 x^{5}} + \frac{1}{705894 x^{6}}$$

Förenkla:

$$\int{\frac{\frac{x}{823543} - \frac{1}{117649}}{x^{7}} d x} = \frac{35 - 6 x}{24706290 x^{6}}$$

Lägg till integrationskonstanten:

$$\int{\frac{\frac{x}{823543} - \frac{1}{117649}}{x^{7}} d x} = \frac{35 - 6 x}{24706290 x^{6}}+C$$

$$$\int \frac{x - 7}{823543 x^{7}}\, dx = \frac{35 - 6 x}{24706290 x^{6}} + C$$$A