Integraal van $$$\frac{\sqrt{2} \left(x^{2} - 25\right)}{x}$$$
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Uw invoer
Bepaal $$$\int \frac{\sqrt{2} \left(x^{2} - 25\right)}{x}\, dx$$$.
Oplossing
Expand the expression:
$${\color{red}{\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x}}} = {\color{red}{\int{\left(\sqrt{2} x - \frac{25 \sqrt{2}}{x}\right)d x}}}$$
Integreer termgewijs:
$${\color{red}{\int{\left(\sqrt{2} x - \frac{25 \sqrt{2}}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{25 \sqrt{2}}{x} d x} + \int{\sqrt{2} x d x}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\sqrt{2}$$$ en $$$f{\left(x \right)} = x$$$:
$$- \int{\frac{25 \sqrt{2}}{x} d x} + {\color{red}{\int{\sqrt{2} x d x}}} = - \int{\frac{25 \sqrt{2}}{x} d x} + {\color{red}{\sqrt{2} \int{x d x}}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:
$$- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\int{x d x}}}=- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{25 \sqrt{2}}{x} d x} + \sqrt{2} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=25 \sqrt{2}$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$\frac{\sqrt{2} x^{2}}{2} - {\color{red}{\int{\frac{25 \sqrt{2}}{x} d x}}} = \frac{\sqrt{2} x^{2}}{2} - {\color{red}{\left(25 \sqrt{2} \int{\frac{1}{x} d x}\right)}}$$
De integraal van $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$$\frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} {\color{red}{\int{\frac{1}{x} d x}}} = \frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
Dus,
$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} x^{2}}{2} - 25 \sqrt{2} \ln{\left(\left|{x}\right| \right)}$$
Vereenvoudig:
$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} \left(x^{2} - 50 \ln{\left(\left|{x}\right| \right)}\right)}{2}$$
Voeg de integratieconstante toe:
$$\int{\frac{\sqrt{2} \left(x^{2} - 25\right)}{x} d x} = \frac{\sqrt{2} \left(x^{2} - 50 \ln{\left(\left|{x}\right| \right)}\right)}{2}+C$$
Antwoord
$$$\int \frac{\sqrt{2} \left(x^{2} - 25\right)}{x}\, dx = \frac{\sqrt{2} \left(x^{2} - 50 \ln\left(\left|{x}\right|\right)\right)}{2} + C$$$A