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Integraal van $$$\frac{1}{x y^{2}}$$$ met betrekking tot $$$x$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \frac{1}{x y^{2}}\, dx$$$.
Oplossing
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{y^{2}}$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:
$${\color{red}{\int{\frac{1}{x y^{2}} d x}}} = {\color{red}{\frac{\int{\frac{1}{x} d x}}{y^{2}}}}$$
De integraal van $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$$\frac{{\color{red}{\int{\frac{1}{x} d x}}}}{y^{2}} = \frac{{\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{y^{2}}$$
Dus,
$$\int{\frac{1}{x y^{2}} d x} = \frac{\ln{\left(\left|{x}\right| \right)}}{y^{2}}$$
Voeg de integratieconstante toe:
$$\int{\frac{1}{x y^{2}} d x} = \frac{\ln{\left(\left|{x}\right| \right)}}{y^{2}}+C$$
Antwoord
$$$\int \frac{1}{x y^{2}}\, dx = \frac{\ln\left(\left|{x}\right|\right)}{y^{2}} + C$$$A