Solve these by integrating, remembering to include the constants of integration, and plugging in the given values to solve for the constants of integration.
----------------------------------------------------------------------------------
For the first problem:
f''(x)=20x3+12x2+6
f'(x)=5x4+4x3+6x+C1
f(x)=x5+x4+3x2+C1x+C2
f(0)=(0)5+(0)4+3(0)2+C1(0)+C2
C2=9
f(x)=x5+x4+3x2+C1x+9
f(1)=(1)5+(1)4+3(1)2+C1(1)+9
7=14+C1
C1=-7
f(x)=x5+x4+3x2-7x+9
----------------------------------------------------------------------------------
For the second problem:
a(t)=cos(t)+sin(t)
v(t)=sin(t)-cos(t)+C1
v(0)=sin(0)-cos(0)+C1
3=0-(1)+C1
C1=4
v(t)=sin(t)-cos(t)+4
s(t)=-cos(t)-sin(t)+4t+C2
s(0)=-cos(0)-sin(0)+4(0)+C2
2=-(1)-0+0+C
2
C2=3
s(t)=-cos(t)-sin(t)+4t+3