Soit la fonction \( f \) définie sur R par : \( f(x)=\frac{4 x-3}{x^{2}+1} \) 1. Montrer que : \( \forall(x, y) \in \mathbb{R}^{2} \) tel que \( x \neq y \) \[ \frac{f(x)-f(y)}{x-y}=\frac{(2 x+1)(2-y)+(2 y+1)(2-x)}{\left(x^{2}+1\right)\left(y^{2}+1\right)} \] 2. Étudier les variations de f sur \( \left[2,+\infty\left[,\left[-\frac{1}{2}, 2[\right.\right.\right. \) et \( \left.]-\infty,-\frac{1}{2}\right] \) 3. Déterminer la valeur minimale absolue de \( f \) 4. montrer que : \( f([2,+\infty[)=] 0,1] \)

9(a) Find \( \int \frac{x+2}{x^{2}+2 x-3} \mathrm{~d} x \) \( \begin{array}{ll}\text { (b) Find } \int 2 x \sin x \mathrm{~d} x\end{array} \) \( \begin{array}{ll}\text { (c) Using the substitution } x=3 \sin \theta \text {, find the exact value of } \int_{0}^{\frac{3}{2}} \sqrt{9-x^{2}} \mathrm{~d} x \\ 9 & \text { (a) } \frac{3}{4} \ln |x-1|+\frac{1}{4} \ln |x+3|+C \text { or } \frac{1}{2} \ln \left|x^{2}+2 x-3\right|+\frac{1}{4} \ln \left|\frac{x-1}{x+3}\right|+C \\ \text { (b) }-2 x \cos x+2 \sin x+C & \text { (c) } \frac{9}{8} \sqrt{3}+\frac{3}{4} \pi\end{array} \) \( \begin{array}{ll}\text { (b) }\end{array} \)

احسب المسافة التي قطعها الجسم المتحرك وفقًا للمعادلة \( s(t) = t^3 - 3t + 2 \) خلال الزمن من \( t = 0 \) إلى \( t = 2 \).

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal mintmum value at \( x=1 \). what is the value of \( a \) and \( c \) ?

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal manamum value at \( x=1 \), + What is the value of a \( f c \) ?

Evaluate the triple integral. \[ \iiint_{G} x y \sin (y z) d V \text {, } \] where \( G \) is the rectangular box defined by the inequalities \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)