Find a function f and a number a such that 2 + ∫(low:a, up: x) (f(t) / t^7)*dt = 5x^-2?

Question

f(x) = ___________
a = ____________

Answer 1

Let g(t) = f(t) / t^7. Then we want to find an a and g such that
∫(a to x) g(t) dt = 5x^(-2) - 2
By the Fundamental Theorem of Calculus we can differentiate to get
g(x) = -10x^(-1)
and therefore ∫(a to x) (-10/t) dt = (5/x^2 - 5/a^2)
so 5/a^2 = 2 => a = √10.
So f(t) = -10t^6 and a = √10.