Integralen av $$$\frac{\sqrt{2} \left(x^{2} - 25\right)}{x}$$$

Bestäm $$$\int \frac{\sqrt{2} \left(x^{2} - 25\right)}{x}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x}}} = {\color{red}{\int{\left(\sqrt{2} x - \frac{25 \sqrt{2}}{x}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(\sqrt{2} x - \frac{25 \sqrt{2}}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{25 \sqrt{2}}{x} d x} + \int{\sqrt{2} x d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\sqrt{2}$$$ och $$$f{\left(x \right)} = x$$$:

$$- \int{\frac{25 \sqrt{2}}{x} d x} + {\color{red}{\int{\sqrt{2} x d x}}} = - \int{\frac{25 \sqrt{2}}{x} d x} + {\color{red}{\sqrt{2} \int{x d x}}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:

$$- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\int{x d x}}}=- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=25 \sqrt{2}$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\frac{\sqrt{2} x^{2}}{2} - {\color{red}{\int{\frac{25 \sqrt{2}}{x} d x}}} = \frac{\sqrt{2} x^{2}}{2} - {\color{red}{\left(25 \sqrt{2} \int{\frac{1}{x} d x}\right)}}$$

Integralen av $$$\frac{1}{x}$$$ är $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$\frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} {\color{red}{\int{\frac{1}{x} d x}}} = \frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Alltså,

$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} \ln{\left(\left|{x}\right| \right)}$$

Förenkla:

$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} \left(x^{2} - 50 \ln{\left(\left|{x}\right| \right)}\right)}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} \left(x^{2} - 50 \ln{\left(\left|{x}\right| \right)}\right)}{2}+C$$

$$$\int \frac{\sqrt{2} \left(x^{2} - 25\right)}{x}\, dx = \frac{\sqrt{2} \left(x^{2} - 50 \ln\left(\left|{x}\right|\right)\right)}{2} + C$$$A