To determine
To sketch and find:
The area represented by g(x) and find the derivative by using Fundamental Theorem of Calculus.
Answer

By using Fundamental Theorem of Calculus part a) g'x=2+sinx
By using Fundamental Theorem of Calculus part b), g'x=2+sinx
Explanation
1) Concept:
The Fundamental Theorem of Calculus-Suppose f is continuous on [a, b] then
a) If gx=∫axftdt, then g'=fx
b) ∫abfxdx=Fb-F(a), F is an Antiderivative of f i.e F'=f
2) Given:
gx=∫0x(2+sint)dt
3) Calculation:
The area represented by gx=∫0x(2+sint)dt is the area between graph of fx=2+sinx and x axis from x = 0 to x = x

Now,
gx=∫0x(2+sint)dt
Take derivative of both sides
g'x=ddx∫0x(2+sint)dt
Use Fundamental Theorem of Calculus part a) with ft=2+sint to get
g'(x)=2+sinx
Therefore, g'(x)=2+sinx
Now by using part b) of Fundamental Theorem of Calculus,
gx=∫0x(2+sint)dt
Here ft=2+sint
The anti derivative of f is Gt=2t-cost
Use Fundamental Theorem of Calculus part b) to get
gx=∫0x(2+sint)dt=Gx-G(0)
Use G(t)=2t-cost so G(x)=2x-cosx and G(0)=2*0-cos(0)
So gx=2x-cosx-(-cos0)
Simplifying
gx=2x-cosx+1
Taking derivative
g'(x)=2(1)-(-sinx)-0
Simplifying
g'(x)=2+sinx
Therefore, g'(x)=2+sinx
Conclusion:
Therefore, by using Fundamental Theorem of Calculus part a) g'x=2+sinx
and by using Fundamental Theorem of Calculus part b), g'x=2+sinx