find f(x) given that f"(x)=sinx, and also f(0)=3, and f(pi/2)=2-(pi/2)

Question

I need to know all steps involved in solving problem

Answer 1

f"(x)=sin(x), f'(x)=-cos(x)+A (A is constant to be determined);

f(x)=-sin(x)+Ax+B (B is constant to be determined).

f(0)=3, so 3=-sin(0)+0+B, making B=3.

f(π/2)=-sin(π/2)+Aπ/2+3=-1+Aπ/2+3=2-π/2,

2+Aπ/2=2-π/2, Aπ/2=-π/2, A=-1.

f(x)=-sin(x)-x+3 or 3-x-sin(x).