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Funktion $$$\sin{\left(x \right)} - \frac{1}{x}$$$ integraali
Aiheeseen liittyvä laskin: Määrättyjen ja epäoleellisten integraalien laskin
Syötteesi
Määritä $$$\int \left(\sin{\left(x \right)} - \frac{1}{x}\right)\, dx$$$.
Ratkaisu
Integroi termi kerrallaan:
$${\color{red}{\int{\left(\sin{\left(x \right)} - \frac{1}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{x} d x} + \int{\sin{\left(x \right)} d x}\right)}}$$
Funktion $$$\frac{1}{x}$$$ integraali on $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{x} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
Sinifunktion integraali on $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$- \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
Näin ollen,
$$\int{\left(\sin{\left(x \right)} - \frac{1}{x}\right)d x} = - \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}$$
Lisää integrointivakio:
$$\int{\left(\sin{\left(x \right)} - \frac{1}{x}\right)d x} = - \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}+C$$
Vastaus
$$$\int \left(\sin{\left(x \right)} - \frac{1}{x}\right)\, dx = \left(- \ln\left(\left|{x}\right|\right) - \cos{\left(x \right)}\right) + C$$$A