Integral dari $$$\ln\left(5 x\right)$$$

Temukan $$$\int \ln\left(5 x\right)\, dx$$$.

Misalkan $$$u=5 x$$$.

Kemudian $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{5}$$$.

Integral tersebut dapat ditulis ulang sebagai

$${\color{red}{\int{\ln{\left(5 x \right)} d x}}} = {\color{red}{\int{\frac{\ln{\left(u \right)}}{5} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{5}$$$ dan $$$f{\left(u \right)} = \ln{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\ln{\left(u \right)}}{5} d u}}} = {\color{red}{\left(\frac{\int{\ln{\left(u \right)} d u}}{5}\right)}}$$

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

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

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

Dengan demikian,

$$\frac{{\color{red}{\int{\ln{\left(u \right)} d u}}}}{5}=\frac{{\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{5}=\frac{{\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{5}$$

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

$$\frac{u \ln{\left(u \right)}}{5} - \frac{{\color{red}{\int{1 d u}}}}{5} = \frac{u \ln{\left(u \right)}}{5} - \frac{{\color{red}{u}}}{5}$$

Ingat bahwa $$$u=5 x$$$:

$$- \frac{{\color{red}{u}}}{5} + \frac{{\color{red}{u}} \ln{\left({\color{red}{u}} \right)}}{5} = - \frac{{\color{red}{\left(5 x\right)}}}{5} + \frac{{\color{red}{\left(5 x\right)}} \ln{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Oleh karena itu,

$$\int{\ln{\left(5 x \right)} d x} = x \ln{\left(5 x \right)} - x$$

Sederhanakan:

$$\int{\ln{\left(5 x \right)} d x} = x \left(\ln{\left(x \right)} - 1 + \ln{\left(5 \right)}\right)$$

Tambahkan konstanta integrasi:

$$\int{\ln{\left(5 x \right)} d x} = x \left(\ln{\left(x \right)} - 1 + \ln{\left(5 \right)}\right)+C$$

$$$\int \ln\left(5 x\right)\, dx = x \left(\ln\left(x\right) - 1 + \ln\left(5\right)\right) + C$$$A