Answer
$f(x)=2x-sinx+6$
Work Step by Step
Given:
$f''(x)=sinx,f'(0)=1,f(0)=6$
Integrate $f''(x)$ to get $f'(x)$.
$f'(x)=\int sinxdx$
$f'(x)=-cosx+C$
Solve for $C$ by setting $f'(x)=1$ and plugging in $0$ for $x$ to get the particular solution for $f'(x)$.
$1=-cos(0)+C$
$1=-1+C$
$C=2$
Thus:
$f'(x)=-cosx+1$
Integrate $f'(x)$ to get $f(x)$.
$f(x)=\int (-cosx+2)dx$
$f(x)=2x-sinx+C$
Find the particular solution again by setting $f(x)=6$ and plugging in $0$ for $x$ to find $C$.
$6=2(0)-sin(0)+C$
$6=0-0+C$
$C=6$
Thus:
$f(x)=2x-sinx+6$
