Integral dari $$$x - \frac{x^{2}}{l^{2}}$$$ terhadap $$$x$$$

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

Integralkan suku demi suku:

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

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

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\frac{1}{l^{2}}$$$ dan $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{x^{2}}{2} - {\color{red}{\int{\frac{x^{2}}{l^{2}} d x}}} = \frac{x^{2}}{2} - {\color{red}{\frac{\int{x^{2} d x}}{l^{2}}}}$$

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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