Integral dari $$$x e^{3} \cos{\left(x e^{3} \right)}$$$

Temukan $$$\int x e^{3} \cos{\left(x e^{3} \right)}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=e^{3}$$$ dan $$$f{\left(x \right)} = x \cos{\left(x e^{3} \right)}$$$:

$${\color{red}{\int{x e^{3} \cos{\left(x e^{3} \right)} d x}}} = {\color{red}{e^{3} \int{x \cos{\left(x e^{3} \right)} d x}}}$$

Untuk integral $$$\int{x \cos{\left(x e^{3} \right)} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=x$$$ dan $$$\operatorname{dv}=\cos{\left(x e^{3} \right)} dx$$$.

Maka $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{\cos{\left(x e^{3} \right)} d x}=\frac{\sin{\left(x e^{3} \right)}}{e^{3}}$$$ (langkah-langkah dapat dilihat di »).

Integral tersebut dapat ditulis ulang sebagai

$$e^{3} {\color{red}{\int{x \cos{\left(x e^{3} \right)} d x}}}=e^{3} {\color{red}{\left(x \cdot \frac{\sin{\left(x e^{3} \right)}}{e^{3}}-\int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} \cdot 1 d x}\right)}}=e^{3} {\color{red}{\left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} d x}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=e^{-3}$$$ dan $$$f{\left(x \right)} = \sin{\left(x e^{3} \right)}$$$:

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - {\color{red}{\int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} d x}}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - {\color{red}{\frac{\int{\sin{\left(x e^{3} \right)} d x}}{e^{3}}}}\right)$$

Misalkan $$$u=x e^{3}$$$.

Kemudian $$$du=\left(x e^{3}\right)^{\prime }dx = e^{3} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{e^{3}}$$$.

Oleh karena itu,

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\sin{\left(x e^{3} \right)} d x}}}}{e^{3}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{e^{3}} d u}}}}{e^{3}}\right)$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=e^{-3}$$$ dan $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{e^{3}} d u}}}}{e^{3}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{e^{3}}}}}{e^{3}}\right)$$

Integral dari sinus adalah $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{e^{6}}\right)$$

Ingat bahwa $$$u=x e^{3}$$$:

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{u}} \right)}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{x e^{3}}} \right)}}{e^{6}}\right)$$

Oleh karena itu,

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left(x e^{3} \right)}}{e^{6}}\right) e^{3}$$

Sederhanakan:

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}$$

Tambahkan konstanta integrasi:

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}+C$$

$$$\int x e^{3} \cos{\left(x e^{3} \right)}\, dx = \left(x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}\right) + C$$$A