Integral dari $$$e^{\sqrt[3]{x}}$$$

Temukan $$$\int e^{\sqrt[3]{x}}\, dx$$$.

Misalkan $$$u=\sqrt[3]{x}$$$.

Kemudian $$$du=\left(\sqrt[3]{x}\right)^{\prime }dx = \frac{1}{3 x^{\frac{2}{3}}} dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\frac{dx}{x^{\frac{2}{3}}} = 3 du$$$.

Oleh karena itu,

$${\color{red}{\int{e^{\sqrt[3]{x}} d x}}} = {\color{red}{\int{3 u^{2} e^{u} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=3$$$ dan $$$f{\left(u \right)} = u^{2} e^{u}$$$:

$${\color{red}{\int{3 u^{2} e^{u} d u}}} = {\color{red}{\left(3 \int{u^{2} e^{u} d u}\right)}}$$

Untuk integral $$$\int{u^{2} e^{u} d u}$$$, gunakan integrasi parsial $$$\int \operatorname{\mu} \operatorname{dv} = \operatorname{\mu}\operatorname{v} - \int \operatorname{v} \operatorname{d\mu}$$$.

Misalkan $$$\operatorname{\mu}=u^{2}$$$ dan $$$\operatorname{dv}=e^{u} du$$$.

Maka $$$\operatorname{d\mu}=\left(u^{2}\right)^{\prime }du=2 u du$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{e^{u} d u}=e^{u}$$$ (langkah-langkah dapat dilihat di »).

Dengan demikian,

$$3 {\color{red}{\int{u^{2} e^{u} d u}}}=3 {\color{red}{\left(u^{2} \cdot e^{u}-\int{e^{u} \cdot 2 u d u}\right)}}=3 {\color{red}{\left(u^{2} e^{u} - \int{2 u e^{u} d u}\right)}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=2$$$ dan $$$f{\left(u \right)} = u e^{u}$$$:

$$3 u^{2} e^{u} - 3 {\color{red}{\int{2 u e^{u} d u}}} = 3 u^{2} e^{u} - 3 {\color{red}{\left(2 \int{u e^{u} d u}\right)}}$$

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

Misalkan $$$\operatorname{\mu}=u$$$ dan $$$\operatorname{dv}=e^{u} du$$$.

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

Integral tersebut dapat ditulis ulang sebagai

$$3 u^{2} e^{u} - 6 {\color{red}{\int{u e^{u} d u}}}=3 u^{2} e^{u} - 6 {\color{red}{\left(u \cdot e^{u}-\int{e^{u} \cdot 1 d u}\right)}}=3 u^{2} e^{u} - 6 {\color{red}{\left(u e^{u} - \int{e^{u} d u}\right)}}$$

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

$$3 u^{2} e^{u} - 6 u e^{u} + 6 {\color{red}{\int{e^{u} d u}}} = 3 u^{2} e^{u} - 6 u e^{u} + 6 {\color{red}{e^{u}}}$$

Ingat bahwa $$$u=\sqrt[3]{x}$$$:

$$6 e^{{\color{red}{u}}} - 6 {\color{red}{u}} e^{{\color{red}{u}}} + 3 {\color{red}{u}}^{2} e^{{\color{red}{u}}} = 6 e^{{\color{red}{\sqrt[3]{x}}}} - 6 {\color{red}{\sqrt[3]{x}}} e^{{\color{red}{\sqrt[3]{x}}}} + 3 {\color{red}{\sqrt[3]{x}}}^{2} e^{{\color{red}{\sqrt[3]{x}}}}$$

Oleh karena itu,

$$\int{e^{\sqrt[3]{x}} d x} = 3 x^{\frac{2}{3}} e^{\sqrt[3]{x}} - 6 \sqrt[3]{x} e^{\sqrt[3]{x}} + 6 e^{\sqrt[3]{x}}$$

Sederhanakan:

$$\int{e^{\sqrt[3]{x}} d x} = 3 \left(x^{\frac{2}{3}} - 2 \sqrt[3]{x} + 2\right) e^{\sqrt[3]{x}}$$

Tambahkan konstanta integrasi:

$$\int{e^{\sqrt[3]{x}} d x} = 3 \left(x^{\frac{2}{3}} - 2 \sqrt[3]{x} + 2\right) e^{\sqrt[3]{x}}+C$$

$$$\int e^{\sqrt[3]{x}}\, dx = 3 \left(x^{\frac{2}{3}} - 2 \sqrt[3]{x} + 2\right) e^{\sqrt[3]{x}} + C$$$A