Intégrale de $$$x \ln\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) \sin{\left(\tanh{\left(\eta \right)} \right)}$$$ par rapport à $$$x$$$

Déterminez $$$\int x \ln\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) \sin{\left(\tanh{\left(\eta \right)} \right)}\, dx$$$.

Soit $$$u=x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$.

Alors $$$du=\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right)^{\prime }dx = 2 x \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} dx$$$ (les étapes peuvent être vues »), et nous obtenons $$$x dx = \frac{du}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$$.

L’intégrale peut être réécrite sous la forme

$${\color{red}{\int{x \ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} \sin{\left(\tanh{\left(\eta \right)} \right)} d x}}} = {\color{red}{\int{\frac{\ln{\left(u \right)} \sin{\left(\tanh{\left(\eta \right)} \right)}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}} d u}}}$$

Appliquez la règle du facteur constant $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ avec $$$c=\frac{\sin{\left(\tanh{\left(\eta \right)} \right)}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$$ et $$$f{\left(u \right)} = \ln{\left(u \right)}$$$ :

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

Pour l’intégrale $$$\int{\ln{\left(u \right)} d u}$$$, utilisez l’intégration par parties $$$\int \operatorname{a} \operatorname{dv} = \operatorname{a}\operatorname{v} - \int \operatorname{v} \operatorname{da}$$$.

Soient $$$\operatorname{a}=\ln{\left(u \right)}$$$ et $$$\operatorname{dv}=du$$$.

Donc $$$\operatorname{da}=\left(\ln{\left(u \right)}\right)^{\prime }du=\frac{du}{u}$$$ (les étapes peuvent être consultées ») et $$$\operatorname{v}=\int{1 d u}=u$$$ (les étapes peuvent être consultées »).

Donc,

$$\frac{\sin{\left(\tanh{\left(\eta \right)} \right)} {\color{red}{\int{\ln{\left(u \right)} d u}}}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}=\frac{\sin{\left(\tanh{\left(\eta \right)} \right)} {\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}=\frac{\sin{\left(\tanh{\left(\eta \right)} \right)} {\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$

Appliquez la règle de la constante $$$\int c\, du = c u$$$ avec $$$c=1$$$:

$$\frac{\sin{\left(\tanh{\left(\eta \right)} \right)} \left(u \ln{\left(u \right)} - {\color{red}{\int{1 d u}}}\right)}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}} = \frac{\sin{\left(\tanh{\left(\eta \right)} \right)} \left(u \ln{\left(u \right)} - {\color{red}{u}}\right)}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$

Rappelons que $$$u=x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$ :

$$\frac{\sin{\left(\tanh{\left(\eta \right)} \right)} \left(- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}\right)}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}} = \frac{\sin{\left(\tanh{\left(\eta \right)} \right)} \left(- {\color{red}{x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}} + {\color{red}{x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}} \ln{\left({\color{red}{x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}} \right)}\right)}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$

Par conséquent,

$$\int{x \ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} \sin{\left(\tanh{\left(\eta \right)} \right)} d x} = \frac{\left(x^{2} \ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} - x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) \sin{\left(\tanh{\left(\eta \right)} \right)}}{2 \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}}$$

Simplifier:

$$\int{x \ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} \sin{\left(\tanh{\left(\eta \right)} \right)} d x} = \frac{x^{2} \left(\ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} - 1\right) \sin{\left(\tanh{\left(\eta \right)} \right)}}{2}$$

Ajouter la constante d'intégration :

$$\int{x \ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} \sin{\left(\tanh{\left(\eta \right)} \right)} d x} = \frac{x^{2} \left(\ln{\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \right)} - 1\right) \sin{\left(\tanh{\left(\eta \right)} \right)}}{2}+C$$

$$$\int x \ln\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) \sin{\left(\tanh{\left(\eta \right)} \right)}\, dx = \frac{x^{2} \left(\ln\left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) - 1\right) \sin{\left(\tanh{\left(\eta \right)} \right)}}{2} + C$$$A