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

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

Skriv om cosinus i termer av sinus med hjälp av formeln $$$\cos\left(x\right)=\sin\left(x + \frac{\pi}{2}\right)$$$ och skriv sedan om sinus med dubbelvinkelformeln $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

$${\color{red}{\int{\frac{1}{2 \cos{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{4 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}}$$

Multiplicera täljare och nämnare med $$$\sec^2\left(\frac{x}{2} + \frac{\pi}{4} \right)$$$:

$${\color{red}{\int{\frac{1}{4 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}} = {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{4 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}}$$

Låt $$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$ vara.

Då $$$du=\left(\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$ (stegen kan ses »), och vi har att $$$\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} dx = 2 du$$$.

Alltså,

$${\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{4 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}} = {\color{red}{\int{\frac{1}{2 u} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{1}{2 u} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{2}\right)}}$$

Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

Kom ihåg att $$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$:

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} = \frac{\ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}}}\right| \right)}}{2}$$

Alltså,

$$\int{\frac{1}{2 \cos{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{2 \cos{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{2}+C$$

$$$\int \frac{1}{2 \cos{\left(x \right)}}\, dx = \frac{\ln\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right|\right)}{2} + C$$$A