sovle the differential equation, basic .:Resolve:.

[voidflux]

hello everyone, I'm having troubles figuring out this problem and i'm trying to study for a test but its an even number and I have no idea what the answer is. THe problem is:

f''(x) = sinx, f'(0) = 1, f(0) = 6;

I understand how to do non-trig ones
f''(x) = 2, f'(2) = 5, f(2) = 10
f'(x) = 2x + c1;
f'(2) = 2(2) + c1;
5 = 4 + c1;
c1 = 1.
f'(x) = 2x+1;
f(x) = x^2+1x+c2;
f(2) = 2^(2)+1(2)+c2;
10 = 6 + c2
c2 = 4;
f(x) = x^2+x+4

I"m getting:
f(x) = -sinx+2x+6
which i htink is right i just want a double check on this

[TheManWhoCan]

f''(x) = sinx

Integrate both sides w.r.t. x:

f'(x) = -cosx + c

Since f'(0) = 1, c = 2

f'(x) = 2 - cosx

Integrate both sides w.r.t. x:

f(x) = 2x - sinx + c

Since f(0) = 6, c =6

f(x) = 2x - sinx + 6

[voidflux]

Thanks!