Funktion $$$\frac{x^{2} - 4}{x}$$$ integraali

Määritä $$$\int \frac{x^{2} - 4}{x}\, dx$$$.

Expand the expression:

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

Integroi termi kerrallaan:

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

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

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=4$$$ ja $$$f{\left(x \right)} = \frac{1}{x}$$$:

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

Funktion $$$\frac{1}{x}$$$ integraali on $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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