Integral von $$$7 \tan^{3}{\left(x \right)} \sec{\left(x \right)}$$$

Bestimme $$$\int 7 \tan^{3}{\left(x \right)} \sec{\left(x \right)}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=7$$$ und $$$f{\left(x \right)} = \tan^{3}{\left(x \right)} \sec{\left(x \right)}$$$ an:

$${\color{red}{\int{7 \tan^{3}{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\left(7 \int{\tan^{3}{\left(x \right)} \sec{\left(x \right)} d x}\right)}}$$

Klammern Sie einen Tangens aus und drücken Sie alles andere in Termen der Sekans aus, unter Verwendung der Formel $$$\tan^2\left(x \right)=\sec^2\left(x \right)-1$$$:

$$7 {\color{red}{\int{\tan^{3}{\left(x \right)} \sec{\left(x \right)} d x}}} = 7 {\color{red}{\int{\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$

Sei $$$u=\sec{\left(x \right)}$$$.

Dann $$$du=\left(\sec{\left(x \right)}\right)^{\prime }dx = \tan{\left(x \right)} \sec{\left(x \right)} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$.

Daher,

$$7 {\color{red}{\int{\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = 7 {\color{red}{\int{\left(u^{2} - 1\right)d u}}}$$

Gliedweise integrieren:

$$7 {\color{red}{\int{\left(u^{2} - 1\right)d u}}} = 7 {\color{red}{\left(- \int{1 d u} + \int{u^{2} d u}\right)}}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=1$$$ an:

$$7 \int{u^{2} d u} - 7 {\color{red}{\int{1 d u}}} = 7 \int{u^{2} d u} - 7 {\color{red}{u}}$$

Wenden Sie die Potenzregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=2$$$ an:

$$- 7 u + 7 {\color{red}{\int{u^{2} d u}}}=- 7 u + 7 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- 7 u + 7 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Zur Erinnerung: $$$u=\sec{\left(x \right)}$$$:

$$- 7 {\color{red}{u}} + \frac{7 {\color{red}{u}}^{3}}{3} = - 7 {\color{red}{\sec{\left(x \right)}}} + \frac{7 {\color{red}{\sec{\left(x \right)}}}^{3}}{3}$$

Daher,

$$\int{7 \tan^{3}{\left(x \right)} \sec{\left(x \right)} d x} = \frac{7 \sec^{3}{\left(x \right)}}{3} - 7 \sec{\left(x \right)}$$

Vereinfachen:

$$\int{7 \tan^{3}{\left(x \right)} \sec{\left(x \right)} d x} = \frac{7 \left(\sec^{2}{\left(x \right)} - 3\right) \sec{\left(x \right)}}{3}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{7 \tan^{3}{\left(x \right)} \sec{\left(x \right)} d x} = \frac{7 \left(\sec^{2}{\left(x \right)} - 3\right) \sec{\left(x \right)}}{3}+C$$

$$$\int 7 \tan^{3}{\left(x \right)} \sec{\left(x \right)}\, dx = \frac{7 \left(\sec^{2}{\left(x \right)} - 3\right) \sec{\left(x \right)}}{3} + C$$$A