Integralen av $$$\sec^{2}{\left(\frac{\pi x}{3} \right)}$$$

Bestäm $$$\int \sec^{2}{\left(\frac{\pi x}{3} \right)}\, dx$$$.

Låt $$$u=\frac{\pi x}{3}$$$ vara.

Då $$$du=\left(\frac{\pi x}{3}\right)^{\prime }dx = \frac{\pi}{3} dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{3 du}{\pi}$$$.

Alltså,

$${\color{red}{\int{\sec^{2}{\left(\frac{\pi x}{3} \right)} d x}}} = {\color{red}{\int{\frac{3 \sec^{2}{\left(u \right)}}{\pi} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{3}{\pi}$$$ och $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{3 \sec^{2}{\left(u \right)}}{\pi} d u}}} = {\color{red}{\left(\frac{3 \int{\sec^{2}{\left(u \right)} d u}}{\pi}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}}}{\pi} = \frac{3 {\color{red}{\tan{\left(u \right)}}}}{\pi}$$

Kom ihåg att $$$u=\frac{\pi x}{3}$$$:

$$\frac{3 \tan{\left({\color{red}{u}} \right)}}{\pi} = \frac{3 \tan{\left({\color{red}{\left(\frac{\pi x}{3}\right)}} \right)}}{\pi}$$

Alltså,

$$\int{\sec^{2}{\left(\frac{\pi x}{3} \right)} d x} = \frac{3 \tan{\left(\frac{\pi x}{3} \right)}}{\pi}$$

Lägg till integrationskonstanten:

$$\int{\sec^{2}{\left(\frac{\pi x}{3} \right)} d x} = \frac{3 \tan{\left(\frac{\pi x}{3} \right)}}{\pi}+C$$

$$$\int \sec^{2}{\left(\frac{\pi x}{3} \right)}\, dx = \frac{3 \tan{\left(\frac{\pi x}{3} \right)}}{\pi} + C$$$A