Given f′(x)=7 cos(x) −11 sin(x) and f(0)=−1, find f(x). f(x)=?

Question

Given f′(x)=7 cos(x)−11 sin(x) and f(0)=−1, find f(x).

f(x)=?

Answer 1

Integrating f(x)= 7 sin x + 11 cos x +c

-1 = 11+c

C=-12

Integrating f(x)= 7 sin x + 11 cos x -12

Answer 2

Integrating, we have f(x) = ∫ f′(x) dx = ∫ 7 cos(x)−11 sin(x) dx

= 7 sin(x) + 11 cos(x) + C

Putting x = 0, we have sin(0) = 0 and cos(0) = 1, so

-1 = f(0) = 11 + C,

so that C = -12

and f(x) = 7 sin(x) + 11 cos(x) - 12