Integral dari $$$\ln^{3}\left(x\right)$$$

Temukan $$$\int \ln^{3}\left(x\right)\, dx$$$.

Untuk integral $$$\int{\ln{\left(x \right)}^{3} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=\ln{\left(x \right)}^{3}$$$ dan $$$\operatorname{dv}=dx$$$.

Maka $$$\operatorname{du}=\left(\ln{\left(x \right)}^{3}\right)^{\prime }dx=\frac{3 \ln{\left(x \right)}^{2}}{x} dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{1 d x}=x$$$ (langkah-langkah dapat dilihat di »).

Integralnya menjadi

$${\color{red}{\int{\ln{\left(x \right)}^{3} d x}}}={\color{red}{\left(\ln{\left(x \right)}^{3} \cdot x-\int{x \cdot \frac{3 \ln{\left(x \right)}^{2}}{x} d x}\right)}}={\color{red}{\left(x \ln{\left(x \right)}^{3} - \int{3 \ln{\left(x \right)}^{2} d x}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=3$$$ dan $$$f{\left(x \right)} = \ln{\left(x \right)}^{2}$$$:

$$x \ln{\left(x \right)}^{3} - {\color{red}{\int{3 \ln{\left(x \right)}^{2} d x}}} = x \ln{\left(x \right)}^{3} - {\color{red}{\left(3 \int{\ln{\left(x \right)}^{2} d x}\right)}}$$

Untuk integral $$$\int{\ln{\left(x \right)}^{2} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ dan $$$\operatorname{dv}=dx$$$.

Maka $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{1 d x}=x$$$ (langkah-langkah dapat dilihat di »).

Integral tersebut dapat ditulis ulang sebagai

$$x \ln{\left(x \right)}^{3} - 3 {\color{red}{\int{\ln{\left(x \right)}^{2} d x}}}=x \ln{\left(x \right)}^{3} - 3 {\color{red}{\left(\ln{\left(x \right)}^{2} \cdot x-\int{x \cdot \frac{2 \ln{\left(x \right)}}{x} d x}\right)}}=x \ln{\left(x \right)}^{3} - 3 {\color{red}{\left(x \ln{\left(x \right)}^{2} - \int{2 \ln{\left(x \right)} d x}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=2$$$ dan $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$$x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 3 {\color{red}{\int{2 \ln{\left(x \right)} d x}}} = x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 3 {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}$$

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

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

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

Jadi,

$$x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 {\color{red}{\int{\ln{\left(x \right)} d x}}}=x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=1$$$:

$$x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 x \ln{\left(x \right)} - 6 {\color{red}{\int{1 d x}}} = x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 x \ln{\left(x \right)} - 6 {\color{red}{x}}$$

Oleh karena itu,

$$\int{\ln{\left(x \right)}^{3} d x} = x \ln{\left(x \right)}^{3} - 3 x \ln{\left(x \right)}^{2} + 6 x \ln{\left(x \right)} - 6 x$$

Sederhanakan:

$$\int{\ln{\left(x \right)}^{3} d x} = x \left(\ln{\left(x \right)}^{3} - 3 \ln{\left(x \right)}^{2} + 6 \ln{\left(x \right)} - 6\right)$$

Tambahkan konstanta integrasi:

$$\int{\ln{\left(x \right)}^{3} d x} = x \left(\ln{\left(x \right)}^{3} - 3 \ln{\left(x \right)}^{2} + 6 \ln{\left(x \right)} - 6\right)+C$$

$$$\int \ln^{3}\left(x\right)\, dx = x \left(\ln^{3}\left(x\right) - 3 \ln^{2}\left(x\right) + 6 \ln\left(x\right) - 6\right) + C$$$A