Integraali $$$x - \frac{x^{2}}{l^{2}}$$$:stä muuttujan $$$x$$$ suhteen

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

Integroi termi kerrallaan:

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

Sovella potenssisääntöä $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=1$$$:

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{1}{l^{2}}$$$ ja $$$f{\left(x \right)} = x^{2}$$$:

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

Sovella potenssisääntöä $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ käyttäen $$$n=2$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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