Integral Calculus Proving

Question

Find a function f and a number c such that:

6+∫_c^x (f(t)/t)dt=2√x for all x>0

Answer 1

Let F(x) = 6 + integral (from c to x) [f(t) / t]dt = 2sqrt(x)

By the Fundamental Theorem of Calculus, F'(x) = f(x) / x = 1 / sqrt(x)

So, f(x) = x / sqrt(x).

Also, F(c) = 6 + integral (from c to c) [f(t)/t]dt = 2sqrt(c)

So, 6 + 0 = 2sqrt(c)

sqrt(c) = 3 Thus, c = 9.