Question
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
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Answer to a math question 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
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To find the derivative G'(x) G(x) = \int_x^{\cos(x)} \sin(t^2 + 2) \,dt
\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
Sincef(t, x) = \sin(t^2 + 2) x f x
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 derivativeG'(x) \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)}
Since
Calculating the derivative:
So, the derivative
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