Integral dari $$$- 4 x^{3} - 3 x^{2} - 3 x$$$

Temukan $$$\int \left(- 4 x^{3} - 3 x^{2} - 3 x\right)\, dx$$$.

Integralkan suku demi suku:

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

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

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

Terapkan aturan pangkat $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=3$$$:

$$- \int{3 x d x} - \int{3 x^{2} d x} - 4 {\color{red}{\int{x^{3} d x}}}=- \int{3 x d x} - \int{3 x^{2} d x} - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \int{3 x d x} - \int{3 x^{2} d x} - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

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

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

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

$$- x^{4} - \int{3 x^{2} d x} - 3 {\color{red}{\int{x d x}}}=- x^{4} - \int{3 x^{2} d x} - 3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- x^{4} - \int{3 x^{2} d x} - 3 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

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

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

$$- x^{4} - \frac{3 x^{2}}{2} - 3 {\color{red}{\int{x^{2} d x}}}=- x^{4} - \frac{3 x^{2}}{2} - 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- x^{4} - \frac{3 x^{2}}{2} - 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Oleh karena itu,

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

Sederhanakan:

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

Tambahkan konstanta integrasi:

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

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