Mathematics

Problem

Define the function

F(x)=\int_{0}^{x} \frac{6t+5}{\sqrt{3t^{2}+5t+2}}\,dt.

Answer each part, presenting your work in a logically ordered sequence. a. Find a closed-form expression for

F(x)

. Be sure to simplify constants whenever possible. b. Using the Fundamental Theorem of Calculus, compute

F'(x)

directly from your result in part (a) and verify that

F'(x)=\dfrac{6x+5}{\sqrt{3x^{2}+5x+2}}

. Describe how each differentiation rule you apply preserves equivalence. c. Solve the equation

F(x)=0

and state all real solutions. Justify the validity of any algebraic manipulations you employ when isolating

x

. (If no real solution exists, present a rigorous argument supporting that conclusion.)