Answer:
F(x)=sin(2)*[tex]x^{2}[/tex]-cos(x) +10x + 8
Step-by-step explanation:
in order to find F(x) you need to integrate two times.
-> First time
F'(x) = sin(2)*x + sin(x) + C,
if F'(0)=10 then F'(0)=sin(2)*(0) +sin(0) +c=10, so c=10.
Replacing C=10
F'(x) = sin(2)*x + sin(x) + 10.
-> Second time
F(x)=sin(2)*[tex]x^{2}[/tex]-cos(x) +10x + [tex]c_{2}[/tex]
if F(0) = 7 then sin(2)*0 - cos(0) + 10*0 +[tex]c_{2}[/tex]=7
so [tex]c_{2}[/tex]=8
Replacing this
F(x)=sin(2)*[tex]x^{2}[/tex]-cos(x) +10x + 8
If f(0)=10 and f'(0)=7 the answer is F(x)=sin(2)*[tex]x^{2}[/tex]-cos(x) +7x + 11
if F''(x)=sin(2x)+cos(x), f(0)=10, f'(0)=7 , the answer is F(x)=-sin(2x)/4 -cos(x) +15x/2+11
