Integraal van $$$\frac{1}{\sec{\left(2 x \right)}}$$$

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

Zij $$$u=2 x$$$.

Dan $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{2}$$$.

Dus,

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = \frac{1}{\sec{\left(u \right)}}$$$:

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

Herschrijf de integraand in termen van de cosinus:

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

De integraal van de cosinus is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

We herinneren eraan dat $$$u=2 x$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{2} = \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

Dus,

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

Voeg de integratieconstante toe:

$$\int{\frac{1}{\sec{\left(2 x \right)}} d x} = \frac{\sin{\left(2 x \right)}}{2}+C$$

$$$\int \frac{1}{\sec{\left(2 x \right)}}\, dx = \frac{\sin{\left(2 x \right)}}{2} + C$$$A