Integralen av $$$2 \sec{\left(x \right)}$$$

Bestäm $$$\int 2 \sec{\left(x \right)}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \sec{\left(x \right)}$$$:

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

Skriv om sekanten som $$$\sec\left(x\right)=\frac{1}{\cos\left(x\right)}$$$:

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

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 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

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

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

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

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,

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

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

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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