**Problem Statement:**Find \( F'(x) \) given that \[ F(x) = \int_{0}^{2x^2} \cos(t^2 + 2) \, dt \]**Explanation:**This problem involves finding the derivative of a function defined by an integral using the Fundamental Theorem of Calculus. The function \( F(x) \) is an integral with a variable upper limit, \( 2x^2 \). To find the derivative \( F'(x) \), apply the following process:1. **Recognize the Form**: The function is of the form \( F(x) = \int_{a}^{g(x)} f(t) \, dt \).2. **Use the Fundamental Theorem of Calculus**: If \( F(x) = \int_{a}^{g(x)} f(t) \, dt \), then the derivative is \( F'(x) = f(g(x)) \cdot g'(x) \).3. **Apply the Derivative**: - Here, \( f(t) = \cos(t^2 + 2) \). - \( g(x) = 2x^2 \), so \( g'(x) = \frac{d}{dx}(2x^2) = 4x \).4. **Substitute and Simplify**: - \( F'(x) = \cos((2x^2)^2 + 2) \cdot 4x \). - Simplify the expression inside the cosine: \( (2x^2)^2 = 4x^4 \). - Therefore, \( F'(x) = \cos(4x^4 + 2) \cdot 4x \).This solution uses calculus principles to handle the differentiation of an integral with a variable upper limit effectively.

Name: **Problem Statement:**Find \( F'(x) \) given that \[ F(x) = \int_{0}^
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Problem Statement:**Find \( F'(x) \) given that \[ F(x) = \int_{0}^{2x^2} \cos(t^2 + 2) \, dt \]**Explanation:**This problem involves finding the derivative of a function defined by an integral using the Fundamental Theorem of Calculus. The functi