Integral dari $$$x + y + z$$$ terhadap $$$x$$$

Temukan $$$\int \left(x + y + z\right)\, dx$$$.

Integralkan suku demi suku:

$${\color{red}{\int{\left(x + y + z\right)d x}}} = {\color{red}{\left(\int{x d x} + \int{y d x} + \int{z d x}\right)}}$$

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

$$\int{y d x} + \int{z d x} + {\color{red}{\int{x d x}}}=\int{y d x} + \int{z d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{y d x} + \int{z d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=y$$$:

$$\frac{x^{2}}{2} + \int{z d x} + {\color{red}{\int{y d x}}} = \frac{x^{2}}{2} + \int{z d x} + {\color{red}{x y}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=z$$$:

$$\frac{x^{2}}{2} + x y + {\color{red}{\int{z d x}}} = \frac{x^{2}}{2} + x y + {\color{red}{x z}}$$

Oleh karena itu,

$$\int{\left(x + y + z\right)d x} = \frac{x^{2}}{2} + x y + x z$$

Sederhanakan:

$$\int{\left(x + y + z\right)d x} = \frac{x \left(x + 2 y + 2 z\right)}{2}$$

Tambahkan konstanta integrasi:

$$\int{\left(x + y + z\right)d x} = \frac{x \left(x + 2 y + 2 z\right)}{2}+C$$

$$$\int \left(x + y + z\right)\, dx = \frac{x \left(x + 2 y + 2 z\right)}{2} + C$$$A