4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)

Application 3: Simplifier les expressions suivantes: \( \begin{array}{ll}A=\ln (\sqrt{3})+\ln 3-\ln 9 & ; \\ C=\ln \left(\frac{1}{3}\right)+\ln \left(\frac{3}{5}\right)+\ln \left(\frac{5}{7}\right)+\ln \left(\frac{7}{9}\right) & ;\end{array} \quad D=\ln (\sqrt[3]{2}-\ln 16 \)

1) Sout \( x>0 \). En appliquant le thécorème des accroissement fines à la fonction \( \ln \) montrer que: \( \exists c \in] x, x+1\left[\frac{1}{c}=\ln \left(1+\frac{1}{x}\right)\right. \) 2) \( M q: \exists c \in] 0, x\left[\right. \) arctan \( (x)=\frac{x}{1+c^{2}} \) 3) En déduire que \( \forall x>0: \) \( \frac{1}{1+x}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x} \)

A solid is formed with triangular cross-sections where each triangle's base is defined by the equation \(y = x^2\) on interval \([0, 2]\). What is the volume of this solid?

кцияның туындысын табыңыз: \( \left\{\begin{array}{l}x=t^{3}+3 t^{2}-5 \\ y=3 t^{2}+6 t-3\end{array}\right. \)

\( \int _{0}^{\frac{\pi }{2}}\frac{1}{2+\cos (x)} d x \)