Integral dari $$$\ln\left(x y\right)$$$ terhadap $$$x$$$

Temukan $$$\int \ln\left(x y\right)\, dx$$$.

Misalkan $$$u=x y$$$.

Kemudian $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{y}$$$.

Oleh karena itu,

$${\color{red}{\int{\ln{\left(x y \right)} d x}}} = {\color{red}{\int{\frac{\ln{\left(u \right)}}{y} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{y}$$$ dan $$$f{\left(u \right)} = \ln{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\ln{\left(u \right)}}{y} d u}}} = {\color{red}{\frac{\int{\ln{\left(u \right)} d u}}{y}}}$$

Untuk integral $$$\int{\ln{\left(u \right)} d u}$$$, gunakan integrasi parsial $$$\int \operatorname{m} \operatorname{dv} = \operatorname{m}\operatorname{v} - \int \operatorname{v} \operatorname{dm}$$$.

Misalkan $$$\operatorname{m}=\ln{\left(u \right)}$$$ dan $$$\operatorname{dv}=du$$$.

Maka $$$\operatorname{dm}=\left(\ln{\left(u \right)}\right)^{\prime }du=\frac{du}{u}$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{1 d u}=u$$$ (langkah-langkah dapat dilihat di »).

Integral tersebut dapat ditulis ulang sebagai

$$\frac{{\color{red}{\int{\ln{\left(u \right)} d u}}}}{y}=\frac{{\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{y}=\frac{{\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{y}$$

Terapkan aturan konstanta $$$\int c\, du = c u$$$ dengan $$$c=1$$$:

$$\frac{u \ln{\left(u \right)} - {\color{red}{\int{1 d u}}}}{y} = \frac{u \ln{\left(u \right)} - {\color{red}{u}}}{y}$$

Ingat bahwa $$$u=x y$$$:

$$\frac{- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}}{y} = \frac{- {\color{red}{x y}} + {\color{red}{x y}} \ln{\left({\color{red}{x y}} \right)}}{y}$$

Oleh karena itu,

$$\int{\ln{\left(x y \right)} d x} = \frac{x y \ln{\left(x y \right)} - x y}{y}$$

Sederhanakan:

$$\int{\ln{\left(x y \right)} d x} = x \left(\ln{\left(x y \right)} - 1\right)$$

Tambahkan konstanta integrasi:

$$\int{\ln{\left(x y \right)} d x} = x \left(\ln{\left(x y \right)} - 1\right)+C$$

$$$\int \ln\left(x y\right)\, dx = x \left(\ln\left(x y\right) - 1\right) + C$$$A