Integral dari $$$x^{- a} \ln\left(n\right)$$$ terhadap $$$x$$$

Temukan $$$\int x^{- a} \ln\left(n\right)\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\ln{\left(n \right)}$$$ dan $$$f{\left(x \right)} = x^{- a}$$$:

$${\color{red}{\int{x^{- a} \ln{\left(n \right)} d x}}} = {\color{red}{\ln{\left(n \right)} \int{x^{- a} d x}}}$$

Terapkan aturan pangkat $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=- a$$$:

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

Oleh karena itu,

$$\int{x^{- a} \ln{\left(n \right)} d x} = \frac{x^{1 - a} \ln{\left(n \right)}}{1 - a}$$

Sederhanakan:

$$\int{x^{- a} \ln{\left(n \right)} d x} = - \frac{x^{1 - a} \ln{\left(n \right)}}{a - 1}$$

Tambahkan konstanta integrasi:

$$\int{x^{- a} \ln{\left(n \right)} d x} = - \frac{x^{1 - a} \ln{\left(n \right)}}{a - 1}+C$$

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