5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin ^{2} \theta \). En déduire la monotonie de \( r \). (c) Etudier les variations de \( t \mapsto r(t) \exp (-\arctan (t)) \) et en déduire que \( 0 \leq r(t) \leq r(0) \exp \) (arctar que \( r \) a une limite strictement positive en \( +\infty \).

(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) \( \frac{9}{8} \sqrt{3}+\frac{3}{4} \pi \)

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

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 \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)