\( \lim _ { z \rightarrow i ^ { 0 } } \frac { d } { d z } ( z + i ) ^ { - 2 } \)

Determine the area bounded by the functions \( y = \sin(x) \) and \( y = \cos(x) \) within the interval \( [0, \frac{\pi}{2}] \).

4) The acceleration of a particle moving in a straight line at a time \( t \) seconds, is given by \( a(t)=t+1 \). Given that \( s=0 \) and \( v=8 \) when \( t=0 \), find formula for: (a) The velocity \( v \) (b) The displacement

\( \int \frac { x \ln x - x } { \ln ^ { 3 } x } \)

\( = \int _ { 0 } \frac { \partial t } { 1 + t ^ { 3 } } \)

25. Given \( f(x)=\left\{\begin{array}{c}x+4 \text { if } x<3 \\ 2 x^{2}-x-1 \text { if } x \geq 3\end{array}\right. \) and \( g(x)=\left\{\begin{array}{c}x \text { if } x<-2 \\ 2 x+5 \text { if }-2 \leq x<5 . \\ x^{2}-1 \text { if } x \geq 5\end{array}\right. \) What must be the value of \( a \) so that \( (f \circ g)(a)=\left(8 a^{2}+40 a+50\right)-2 a-6 \) ? A. \( a \) is a value greater than 5. B. \( a \) is any value greater or equal to 3. C. \( a \) is a value between 3 and 5. D. \( a \) is a value smaller than 3. E. \( a \) is a value smaller than -2.