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

Temukan $$$\int \ln\left(u\right)\, du$$$.

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

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

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

Oleh karena itu,

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

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

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

Oleh karena itu,

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

Sederhanakan:

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

Tambahkan konstanta integrasi:

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

$$$\int \ln\left(u\right)\, du = u \left(\ln\left(u\right) - 1\right) + C$$$A