7. On se propose dans cette question de montrer que la fonction \( t \mapsto \theta(t)+t \) tend vers ur réelle quand \( t \rightarrow+\infty \). (a) Etablir que que \( \forall t \geq 0,\left|\theta^{\prime}(t)+1\right| \leq \frac{1}{1+t^{2}} \). En déduire que : i. La fonction \( t \mapsto \theta(t)+t \) est majorée sur \( \mathbb{R}^{+} \).

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

\( \iint_{R} \sqrt{x^{2}+y^{2}} d x d y \) बत्र मान निर्

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

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 \( f c \) ?

Compute the average value of \( f(t) = \sin(t) + \cos(t) \) over the interval \([0, \pi]\).