Integral dari $$$\ln\left(x^{2}\right)$$$

Temukan $$$\int \ln\left(x^{2}\right)\, dx$$$.

Masukan ditulis ulang: $$$\int{\ln{\left(x^{2} \right)} d x}=\int{2 \ln{\left(x \right)} d x}$$$.

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

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

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 »).

Integralnya menjadi

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

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

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

Oleh karena itu,

$$\int{2 \ln{\left(x \right)} d x} = 2 x \ln{\left(x \right)} - 2 x$$

Sederhanakan:

$$\int{2 \ln{\left(x \right)} d x} = 2 x \left(\ln{\left(x \right)} - 1\right)$$

Tambahkan konstanta integrasi:

$$\int{2 \ln{\left(x \right)} d x} = 2 x \left(\ln{\left(x \right)} - 1\right)+C$$

$$$\int \ln\left(x^{2}\right)\, dx = 2 x \left(\ln\left(x\right) - 1\right) + C$$$A