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

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

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

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

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

Dengan demikian,

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

Terapkan aturan konstanta $$$\int c\, dy = c y$$$ dengan $$$c=1$$$:

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

Oleh karena itu,

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

Sederhanakan:

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

Tambahkan konstanta integrasi:

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

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