A particle's velocity function is given by \( v(t) = 3t^{2} - 4t + 2 \). Find the position function if the initial position at time t=0 is 5.

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

\( \iint_{R} \sqrt{x^{2}+y^{2}} d x d y \) 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 \([0, \pi]\).