Funktion $$$\frac{\ln\left(x\right)}{x^{9}}$$$ integraali

Määritä $$$\int \frac{\ln\left(x\right)}{x^{9}}\, dx$$$.

Integraalin $$$\int{\frac{\ln{\left(x \right)}}{x^{9}} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Olkoon $$$\operatorname{u}=\ln{\left(x \right)}$$$ ja $$$\operatorname{dv}=\frac{dx}{x^{9}}$$$.

Tällöin $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{\frac{1}{x^{9}} d x}=- \frac{1}{8 x^{8}}$$$ (vaiheet ovat nähtävissä »).

Näin ollen,

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{9}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{8 x^{8}}\right)-\int{\left(- \frac{1}{8 x^{8}}\right) \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(- \int{\left(- \frac{1}{8 x^{9}}\right)d x} - \frac{\ln{\left(x \right)}}{8 x^{8}}\right)}}$$

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=- \frac{1}{8}$$$ ja $$$f{\left(x \right)} = \frac{1}{x^{9}}$$$:

$$- {\color{red}{\int{\left(- \frac{1}{8 x^{9}}\right)d x}}} - \frac{\ln{\left(x \right)}}{8 x^{8}} = - {\color{red}{\left(- \frac{\int{\frac{1}{x^{9}} d x}}{8}\right)}} - \frac{\ln{\left(x \right)}}{8 x^{8}}$$

Sovella potenssisääntöä $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=-9$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{9}} d x}}}}{8} - \frac{\ln{\left(x \right)}}{8 x^{8}}=\frac{{\color{red}{\int{x^{-9} d x}}}}{8} - \frac{\ln{\left(x \right)}}{8 x^{8}}=\frac{{\color{red}{\frac{x^{-9 + 1}}{-9 + 1}}}}{8} - \frac{\ln{\left(x \right)}}{8 x^{8}}=\frac{{\color{red}{\left(- \frac{x^{-8}}{8}\right)}}}{8} - \frac{\ln{\left(x \right)}}{8 x^{8}}=\frac{{\color{red}{\left(- \frac{1}{8 x^{8}}\right)}}}{8} - \frac{\ln{\left(x \right)}}{8 x^{8}}$$

Näin ollen,

$$\int{\frac{\ln{\left(x \right)}}{x^{9}} d x} = - \frac{\ln{\left(x \right)}}{8 x^{8}} - \frac{1}{64 x^{8}}$$

Sievennä:

$$\int{\frac{\ln{\left(x \right)}}{x^{9}} d x} = \frac{- 8 \ln{\left(x \right)} - 1}{64 x^{8}}$$

Lisää integrointivakio:

$$\int{\frac{\ln{\left(x \right)}}{x^{9}} d x} = \frac{- 8 \ln{\left(x \right)} - 1}{64 x^{8}}+C$$

$$$\int \frac{\ln\left(x\right)}{x^{9}}\, dx = \frac{- 8 \ln\left(x\right) - 1}{64 x^{8}} + C$$$A