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

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

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

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

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

Dengan demikian,

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

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

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

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

Jadi,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=-1$$$ dan $$$f{\left(x \right)} = \sin{\left(\ln{\left(x \right)} \right)}$$$:

$$x \sin{\left(\ln{\left(x \right)} \right)} - x \cos{\left(\ln{\left(x \right)} \right)} + {\color{red}{\int{\left(- \sin{\left(\ln{\left(x \right)} \right)}\right)d x}}} = x \sin{\left(\ln{\left(x \right)} \right)} - x \cos{\left(\ln{\left(x \right)} \right)} + {\color{red}{\left(- \int{\sin{\left(\ln{\left(x \right)} \right)} d x}\right)}}$$

Kita telah sampai pada integral yang sudah pernah kita lihat.

Dengan demikian, kita telah memperoleh persamaan sederhana berikut sehubungan dengan integral:

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

Dengan menyelesaikannya, kita memperoleh bahwa

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

Oleh karena itu,

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

Sederhanakan:

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = - \frac{\sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}$$

Tambahkan konstanta integrasi:

$$\int{\sin{\left(\ln{\left(x \right)} \right)} d x} = - \frac{\sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+C$$

$$$\int \sin{\left(\ln\left(x\right) \right)}\, dx = - \frac{\sqrt{2} x \cos{\left(\ln\left(x\right) + \frac{\pi}{4} \right)}}{2} + C$$$A