Suppose that a particular function f satisfies f"(x)=sin(x)+2cos(x), f(0)=1, f'(0)=1 then we must have f(x)=?

Answer 1

take the integral of f'' to obtain f':

f'(x) = -cos(x) + 2sin(x) + c

We were given f'(0) = 1, so we can use this to find c:

1 = -cos(0) + 2sin(0) + c

1 = -1 + c

c = 2

We can now write f'(x):

f'(x) = -cos(x) + 2sin(x) + 2

To get f(x), take the integral of f'(x):

f(x) = -sin(x) - 2cos(x) + 2x + c

We were given f(0) = 1, so we can use this to find c:

1 = -sin(0) - 2cos(0) + 2*(0) + c

1 = -2 + c

c = 3

We can now write f(x):

f(x) = -sin(x) - 2cos(x) + 2x + 3