Question

F''''(x) = sinx
F'''(-pi/2)=0
F''(0)=0
F'(pi/2)=0
F(-2012pi)=2012
Find F(x):

Answer 1

keep integrating

Answer 2

if F''''=sin(x) then F''' would be -cos(x)+c. Since you have F'''(-pi/2)=0 though you need to solve for c:
0=-cos(-pi/2)+c
c=0
good, we can now say that F'''=-cos(x)
do the same thing with F'' then (F''=-sin(x)+c, 0=-sin(0)+c, c=0, etc.)