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

The equation of a curve is \( y=\frac{75}{(2 x+3)^{2}} \) (i) A point \( P \) moves along the curve in such a way that the \( x \)-coordinate of \( P \) increases at a constant rate of 0.04 units per second. Find the \( x \)-coordinate of \( P \) at the instant the \( y \)-coordinate is decreasing at a rate of 0.2 units per second.

1. Soient (4points) \( I_{n}=\int_{0}^{\pi} \sin ^{n} x d x \) et \( J_{n}=\int_{0}^{\frac{\pi}{2}} \cos ^{n} x d x \) pour \( n \in \mathrm{~N}-\{0\} \) a. Déduireles valeurs de \( : I_{3}, I_{5}, J_{3}, J_{5} \) b. Calculer \( I_{3}-J_{3}, I_{5}+J_{5}, J_{3}+I_{5} J_{5}+I_{3} \) 11. Résoudre les problèmes de Cauchy suivants (6points): a) \( \left\{\begin{array}{l}y^{\prime \prime}-2 y^{\prime}+4 y=x e^{x} \\ y(0)=-1 \text { et } y^{\prime}(0)=2\end{array}\right. \) b) \( \left\{\begin{array}{l}y^{\prime \prime}+4 y^{\prime}+3 y=3 x^{2}-3 \\ y(0)=1 \text { el } y^{\prime}(0)=2\end{array}\right. \)

Calcula \( \int \frac{1}{1+x^{9}} d x \) Sugerencia recuerda que \( \sum_{n=0}^{\infty} z^{n}=\frac{1}{1-z} \)

1. Які вирази не е первісною для заданої функції \( f(x)=\sin 2 x \) ? \[ \begin{array}{ll}F(x)=4-\frac{1}{2} \cos 2 x & F(x)=-\frac{1}{2} \cos 2 x \\ \square \text { Baplär 1 } & F(x)=2-\frac{1}{2} \cos 2 x\end{array} \]

a) \( \left\{\begin{array}{l}y^{\prime \prime}-2 y^{\prime}+4 y=x e^{x} \\ y(0)=-1 \text { et } y^{\prime}(0)=2\end{array}\right. \) b) \( \left\{\begin{array}{l}y^{\prime \prime}+4 y^{\prime}+3 y=3 x^{2}-3 \\ y(0)=1 \text { et } y^{\prime}(0)=2\end{array}\right. \)