Integralen av $$$\frac{\ln\left(x\right)}{x^{24}}$$$

Bestäm $$$\int \frac{\ln\left(x\right)}{x^{24}}\, dx$$$.

För integralen $$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=\ln{\left(x \right)}$$$ och $$$\operatorname{dv}=\frac{dx}{x^{24}}$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\frac{1}{x^{24}} d x}=- \frac{1}{23 x^{23}}$$$ (stegen kan ses »).

Alltså,

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{24}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{23 x^{23}}\right)-\int{\left(- \frac{1}{23 x^{23}}\right) \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(- \int{\left(- \frac{1}{23 x^{24}}\right)d x} - \frac{\ln{\left(x \right)}}{23 x^{23}}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{1}{23}$$$ och $$$f{\left(x \right)} = \frac{1}{x^{24}}$$$:

$$- {\color{red}{\int{\left(- \frac{1}{23 x^{24}}\right)d x}}} - \frac{\ln{\left(x \right)}}{23 x^{23}} = - {\color{red}{\left(- \frac{\int{\frac{1}{x^{24}} d x}}{23}\right)}} - \frac{\ln{\left(x \right)}}{23 x^{23}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-24$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{24}} d x}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\int{x^{-24} d x}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\frac{x^{-24 + 1}}{-24 + 1}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\left(- \frac{x^{-23}}{23}\right)}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\left(- \frac{1}{23 x^{23}}\right)}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}$$

Alltså,

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = - \frac{\ln{\left(x \right)}}{23 x^{23}} - \frac{1}{529 x^{23}}$$

Förenkla:

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = \frac{- 23 \ln{\left(x \right)} - 1}{529 x^{23}}$$

Lägg till integrationskonstanten:

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = \frac{- 23 \ln{\left(x \right)} - 1}{529 x^{23}}+C$$

$$$\int \frac{\ln\left(x\right)}{x^{24}}\, dx = \frac{- 23 \ln\left(x\right) - 1}{529 x^{23}} + C$$$A