f'(t) = ∫f"(t)*dt = ∫(4*e^t + 8*sint)dt = 4*e^t - 8*cost + C1
f(t) = ∫f'(t)*dt = ∫(4*e^t - 8*cost + C1)*dt = 4*e^t - 8*sint + C1*t + C2
當 t = 0時,f(0) = 4*e^0 - 8*sin0 + C1*0 + C2 = 4 + C2 = 0,則 C2 =-4
當 t = π時,f(π) = 4*e^π - 8*sinπ + C1*π + C2 = 4*e^π + C1*π - 4 =0,則 C1= (4 - 4*e^π)/π
所以,f(t) = 4*e^t - 8*sint + (4-4*e^π)*t/π - 4

f"(x) =∫f'"(x)dx = sinx + C1,當 x = 0 時,f"(0) = 0 + C1=6,則 C1=6,f"(x) = sinx + 6
f'(x) = ∫f"(x)dx = -cosx + 6x + C2,當 x =0 時,f'(0) = -1 + 6*0 + C2 =8,則 C2 = 9
f'(x) = -cosx + 6x + 9
f(x) =∫f'(x)dx = -sinx + 3x^2 + 9x + C3,當 x= 0時,f(0)= -0 + 3*0 + 9*0 + C3 = 9,則 C3=9
f(x) = -sinx + 3x^2 + 9x + 9