Tricky Integration substitution problem

Question

The problem is:

$$ 1+ \int_a^x f(t)t^{-6}dt\,=5x^{-1}, x>0 $$

f(x) = ? a = ?

Can someone help me with this problem?

Answer 1

Use the fundamental theorem of calculus. Differentiate both sides wrt $x$:

$f(x)x^{-6} = -5x^{-2}$

$f(x) = -5x^4$

Now put that back into the original equation, which becomes:

$$1 + \int_a^x -\frac{5}{t^2}dt = \frac 5x$$

Solving, we get $\displaystyle 1 -\frac 5a = 0$

$a = 5$