Integral dari $$$x \left(2 x^{5} - 3 x\right) e^{3}$$$

Temukan $$$\int x \left(2 x^{5} - 3 x\right) e^{3}\, dx$$$.

Sederhanakan integran:

$${\color{red}{\int{x \left(2 x^{5} - 3 x\right) e^{3} d x}}} = {\color{red}{\int{x^{2} \left(2 x^{4} - 3\right) e^{3} d x}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=e^{3}$$$ dan $$$f{\left(x \right)} = x^{2} \left(2 x^{4} - 3\right)$$$:

$${\color{red}{\int{x^{2} \left(2 x^{4} - 3\right) e^{3} d x}}} = {\color{red}{e^{3} \int{x^{2} \left(2 x^{4} - 3\right) d x}}}$$

Expand the expression:

$$e^{3} {\color{red}{\int{x^{2} \left(2 x^{4} - 3\right) d x}}} = e^{3} {\color{red}{\int{\left(2 x^{6} - 3 x^{2}\right)d x}}}$$

Integralkan suku demi suku:

$$e^{3} {\color{red}{\int{\left(2 x^{6} - 3 x^{2}\right)d x}}} = e^{3} {\color{red}{\left(- \int{3 x^{2} d x} + \int{2 x^{6} d x}\right)}}$$

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

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

Terapkan aturan pangkat $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=2$$$:

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=2$$$ dan $$$f{\left(x \right)} = x^{6}$$$:

$$e^{3} \left(- x^{3} + {\color{red}{\int{2 x^{6} d x}}}\right) = e^{3} \left(- x^{3} + {\color{red}{\left(2 \int{x^{6} d x}\right)}}\right)$$

Terapkan aturan pangkat $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=6$$$:

$$e^{3} \left(- x^{3} + 2 {\color{red}{\int{x^{6} d x}}}\right)=e^{3} \left(- x^{3} + 2 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}\right)=e^{3} \left(- x^{3} + 2 {\color{red}{\left(\frac{x^{7}}{7}\right)}}\right)$$

Oleh karena itu,

$$\int{x \left(2 x^{5} - 3 x\right) e^{3} d x} = \left(\frac{2 x^{7}}{7} - x^{3}\right) e^{3}$$

Sederhanakan:

$$\int{x \left(2 x^{5} - 3 x\right) e^{3} d x} = \frac{x^{3} \left(2 x^{4} - 7\right) e^{3}}{7}$$

Tambahkan konstanta integrasi:

$$\int{x \left(2 x^{5} - 3 x\right) e^{3} d x} = \frac{x^{3} \left(2 x^{4} - 7\right) e^{3}}{7}+C$$

$$$\int x \left(2 x^{5} - 3 x\right) e^{3}\, dx = \frac{x^{3} \left(2 x^{4} - 7\right) e^{3}}{7} + C$$$A