Integrals Problem

Question

Suppose f''(x)= -25sin(5x) and f'(0)=-5 and f(0)=6

f(pi/4)=

Answer 1

f'(x) = ∫f"(x)dx = -25∫ sin(5x)dx = 5cos(5x) + C

Since f'(0) = -5, 5cos0 + C = -5 So, C = -10

f'(x) = 5cos(5x) - 10

f(x) = ∫f'(x)dx = sin(5x) - 10x + C₁

Since f(0) = 6, sin0 - 0 + C₁ = 6 So, C₁ = 6

Therefore, f(x) = sin(5x) - 10x + 6

f(π/4) = sin(5π/4) - 5π/2 + 6 = -√2/2 - 5π/2 + 6