Develop the selected exercises by deriving G'(x) from the following functions. Apply the following Integration Theorem in each exercise: F(x)=∫_x^(cos(x)) sin(t^2+2)dt

To find the derivative G'(x) of the function G(x) = \int_x^{\cos(x)} \sin(t^2 + 2) \,dt, we'll use the Leibniz Rule for differentiation under the integral sign:

\frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t, x) \, dt \right) = f(b(x), x) \cdot b'(x) - f(a(x), x) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(t, x) \, dt

Since f(t, x) = \sin(t^2 + 2) does not depend on x, the partial derivative of f with respect to x inside the integral is zero, simplifying the formula to:

G'(x) = \sin(\cos^2(x) + 2) \cdot \frac{d}{dx}[\cos(x)] - \sin(x^2 + 2) \cdot \frac{d}{dx}[x]

Calculating the derivative:

G'(x) = \sin(x) \cdot \sin(\cos^2(x) + 2) - \sin(x^2 + 2)

So, the derivative G'(x) is given by \sin(x) \cdot \sin(\cos^2(x) + 2) - \sin(x^2 + 2).

\boxed{G'(x) = \sin(x) \cdot \sin(\cos^2(x) + 2) - \sin(x^2 + 2)}