I need some help on an infinite integral problem?

Question

What is given is that the function is continous from (2,infinity)
f(4)= (-6)
abs[f(x)]< (x^7)+2
integral of f(x)e^(-x/7)dx from 4 to infinity is -4

What is the integral of f `(x)e^(-x/7)dx from 4 to infinity equal to?

Answer 1

Try integration by parts:
∫f'(x)e^(-x/7) dx = f(x)e^(-x/7) - ∫f(x)e^(-x/7) (-1/7) dx
So ∫(4 to ∞) f'(x)e^(-x/7)
= [f(x)e^(-x/7)] [4 to ∞] + 1/7 ∫(4 to ∞) f(x)e^(-x/7) dx
Now we know f(x)e^(-x/7) -> 0 as x -> ∞, both because we are told that the indefinite integral is -4 (to be any real number the integrand must vanish) and because we're given an explicit upper bound on f(x). (Note that e^(-x/7) will dominate any polynomial as x-> ∞.) So we get
= [0 - f(4)e^(-4/7)] + 1/7 (-4)
= -4/7 + 6 e^(-4/7).