Integral von $$$\frac{\tan{\left(\ln\left(x\right) \right)}}{x}$$$

Bestimme $$$\int \frac{\tan{\left(\ln\left(x\right) \right)}}{x}\, dx$$$.

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

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

Das Integral lässt sich umschreiben als

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

Schreibe die Tangente als $$$\tan\left( u \right)=\frac{\sin\left( u \right)}{\cos\left( u \right)}$$$ um:

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

Sei $$$v=\cos{\left(u \right)}$$$.

Dann $$$dv=\left(\cos{\left(u \right)}\right)^{\prime }du = - \sin{\left(u \right)} du$$$ (die Schritte sind » zu sehen), und es gilt $$$\sin{\left(u \right)} du = - dv$$$.

Also,

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} d u}}} = {\color{red}{\int{\left(- \frac{1}{v}\right)d v}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ mit $$$c=-1$$$ und $$$f{\left(v \right)} = \frac{1}{v}$$$ an:

$${\color{red}{\int{\left(- \frac{1}{v}\right)d v}}} = {\color{red}{\left(- \int{\frac{1}{v} d v}\right)}}$$

Das Integral von $$$\frac{1}{v}$$$ ist $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$- {\color{red}{\int{\frac{1}{v} d v}}} = - {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$

Zur Erinnerung: $$$v=\cos{\left(u \right)}$$$:

$$- \ln{\left(\left|{{\color{red}{v}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\cos{\left(u \right)}}}}\right| \right)}$$

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

$$- \ln{\left(\left|{\cos{\left({\color{red}{u}} \right)}}\right| \right)} = - \ln{\left(\left|{\cos{\left({\color{red}{\ln{\left(x \right)}}} \right)}}\right| \right)}$$

Daher,

$$\int{\frac{\tan{\left(\ln{\left(x \right)} \right)}}{x} d x} = - \ln{\left(\left|{\cos{\left(\ln{\left(x \right)} \right)}}\right| \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{\tan{\left(\ln{\left(x \right)} \right)}}{x} d x} = - \ln{\left(\left|{\cos{\left(\ln{\left(x \right)} \right)}}\right| \right)}+C$$

$$$\int \frac{\tan{\left(\ln\left(x\right) \right)}}{x}\, dx = - \ln\left(\left|{\cos{\left(\ln\left(x\right) \right)}}\right|\right) + C$$$A