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Funktion $$$\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}$$$ integraali
Aiheeseen liittyvä laskin: Määrättyjen ja epäoleellisten integraalien laskin
Syötteesi
Määritä $$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$.
Ratkaisu
Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\sqrt{2}$$$ ja $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$:
$${\color{red}{\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\sqrt{2} \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$
Funktion $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ integraali on $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:
$$\sqrt{2} {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = \sqrt{2} {\color{red}{\sec{\left(x \right)}}}$$
Näin ollen,
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}$$
Lisää integrointivakio:
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}+C$$
Vastaus
$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx = \sqrt{2} \sec{\left(x \right)} + C$$$A