Integralen av $$$\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}$$$

Bestäm $$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx$$$.

Låt $$$u=7 x$$$ vara.

Då $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{7}$$$.

Alltså,

$${\color{red}{\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x}}} = {\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{7}$$$ och $$$f{\left(u \right)} = \tan{\left(u \right)} \sec^{5}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\tan{\left(u \right)} \sec^{5}{\left(u \right)}}{7} d u}}} = {\color{red}{\left(\frac{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}{7}\right)}}$$

Låt $$$v=\sec{\left(u \right)}$$$ vara.

Då $$$dv=\left(\sec{\left(u \right)}\right)^{\prime }du = \tan{\left(u \right)} \sec{\left(u \right)} du$$$ (stegen kan ses »), och vi har att $$$\tan{\left(u \right)} \sec{\left(u \right)} du = dv$$$.

Alltså,

$$\frac{{\color{red}{\int{\tan{\left(u \right)} \sec^{5}{\left(u \right)} d u}}}}{7} = \frac{{\color{red}{\int{v^{4} d v}}}}{7}$$

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=4$$$:

$$\frac{{\color{red}{\int{v^{4} d v}}}}{7}=\frac{{\color{red}{\frac{v^{1 + 4}}{1 + 4}}}}{7}=\frac{{\color{red}{\left(\frac{v^{5}}{5}\right)}}}{7}$$

Kom ihåg att $$$v=\sec{\left(u \right)}$$$:

$$\frac{{\color{red}{v}}^{5}}{35} = \frac{{\color{red}{\sec{\left(u \right)}}}^{5}}{35}$$

Kom ihåg att $$$u=7 x$$$:

$$\frac{\sec^{5}{\left({\color{red}{u}} \right)}}{35} = \frac{\sec^{5}{\left({\color{red}{\left(7 x\right)}} \right)}}{35}$$

Alltså,

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}$$

Lägg till integrationskonstanten:

$$\int{\tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)} d x} = \frac{\sec^{5}{\left(7 x \right)}}{35}+C$$

$$$\int \tan{\left(7 x \right)} \sec^{5}{\left(7 x \right)}\, dx = \frac{\sec^{5}{\left(7 x \right)}}{35} + C$$$A