Integralen av $$$\frac{1}{1849 \cos^{2}{\left(x \right)}}$$$

Bestäm $$$\int \frac{1}{1849 \cos^{2}{\left(x \right)}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{1849}$$$ och $$$f{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$$$:

$${\color{red}{\int{\frac{1}{1849 \cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}{1849}\right)}}$$

Skriv om integranden i termer av sekansen:

$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}}}{1849} = \frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{1849}$$

Integralen av $$$\sec^{2}{\left(x \right)}$$$ är $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$$\frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{1849} = \frac{{\color{red}{\tan{\left(x \right)}}}}{1849}$$

Alltså,

$$\int{\frac{1}{1849 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{1849}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{1849 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{1849}+C$$

$$$\int \frac{1}{1849 \cos^{2}{\left(x \right)}}\, dx = \frac{\tan{\left(x \right)}}{1849} + C$$$A