Integral dari $$$x e^{\frac{x}{5}}$$$

Temukan $$$\int x e^{\frac{x}{5}}\, dx$$$.

Untuk integral $$$\int{x e^{\frac{x}{5}} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=x$$$ dan $$$\operatorname{dv}=e^{\frac{x}{5}} dx$$$.

Maka $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{e^{\frac{x}{5}} d x}=5 e^{\frac{x}{5}}$$$ (langkah-langkah dapat dilihat di »).

Jadi,

$${\color{red}{\int{x e^{\frac{x}{5}} d x}}}={\color{red}{\left(x \cdot 5 e^{\frac{x}{5}}-\int{5 e^{\frac{x}{5}} \cdot 1 d x}\right)}}={\color{red}{\left(5 x e^{\frac{x}{5}} - \int{5 e^{\frac{x}{5}} d x}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=5$$$ dan $$$f{\left(x \right)} = e^{\frac{x}{5}}$$$:

$$5 x e^{\frac{x}{5}} - {\color{red}{\int{5 e^{\frac{x}{5}} d x}}} = 5 x e^{\frac{x}{5}} - {\color{red}{\left(5 \int{e^{\frac{x}{5}} d x}\right)}}$$

Misalkan $$$u=\frac{x}{5}$$$.

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

Integral tersebut dapat ditulis ulang sebagai

$$5 x e^{\frac{x}{5}} - 5 {\color{red}{\int{e^{\frac{x}{5}} d x}}} = 5 x e^{\frac{x}{5}} - 5 {\color{red}{\int{5 e^{u} d u}}}$$

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

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

Integral dari fungsi eksponensial adalah $$$\int{e^{u} d u} = e^{u}$$$:

$$5 x e^{\frac{x}{5}} - 25 {\color{red}{\int{e^{u} d u}}} = 5 x e^{\frac{x}{5}} - 25 {\color{red}{e^{u}}}$$

Ingat bahwa $$$u=\frac{x}{5}$$$:

$$5 x e^{\frac{x}{5}} - 25 e^{{\color{red}{u}}} = 5 x e^{\frac{x}{5}} - 25 e^{{\color{red}{\left(\frac{x}{5}\right)}}}$$

Oleh karena itu,

$$\int{x e^{\frac{x}{5}} d x} = 5 x e^{\frac{x}{5}} - 25 e^{\frac{x}{5}}$$

Sederhanakan:

$$\int{x e^{\frac{x}{5}} d x} = 5 \left(x - 5\right) e^{\frac{x}{5}}$$

Tambahkan konstanta integrasi:

$$\int{x e^{\frac{x}{5}} d x} = 5 \left(x - 5\right) e^{\frac{x}{5}}+C$$

$$$\int x e^{\frac{x}{5}}\, dx = 5 \left(x - 5\right) e^{\frac{x}{5}} + C$$$A