Integraal van $$$\tan{\left(4 x \right)} \sec{\left(4 x \right)}$$$

Bepaal $$$\int \tan{\left(4 x \right)} \sec{\left(4 x \right)}\, dx$$$.

Zij $$$u=4 x$$$.

Dan $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{4}$$$.

Dus,

$${\color{red}{\int{\tan{\left(4 x \right)} \sec{\left(4 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)} \sec{\left(u \right)}}{4} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{4}$$$ en $$$f{\left(u \right)} = \tan{\left(u \right)} \sec{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\tan{\left(u \right)} \sec{\left(u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} \sec{\left(u \right)} d u}}{4}\right)}}$$

De integraal van $$$\tan{\left(u \right)} \sec{\left(u \right)}$$$ is $$$\int{\tan{\left(u \right)} \sec{\left(u \right)} d u} = \sec{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\tan{\left(u \right)} \sec{\left(u \right)} d u}}}}{4} = \frac{{\color{red}{\sec{\left(u \right)}}}}{4}$$

We herinneren eraan dat $$$u=4 x$$$:

$$\frac{\sec{\left({\color{red}{u}} \right)}}{4} = \frac{\sec{\left({\color{red}{\left(4 x\right)}} \right)}}{4}$$

Dus,

$$\int{\tan{\left(4 x \right)} \sec{\left(4 x \right)} d x} = \frac{\sec{\left(4 x \right)}}{4}$$

Voeg de integratieconstante toe:

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

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