Hello, and welcome to FMH!
I would begin with:
[MATH]\int_4^{\infty} f(x)e^{-\frac{x}{7}}\,dx=-5[/MATH]
Using integration by parts, where:
[MATH]u=f(x)\implies du=f'(x)\,dx[/MATH]
[MATH]dv=e^{-\frac{x}{7}}\,dx\implies v=-7e^{-\frac{x}{7}}[/MATH]
We may then write:
[MATH]\left[-7f(x)e^{-\frac{x}{7}}\right]_4^{\infty}+7\int_4^{\infty} f'(x)e^{-\frac{x}{7}}\,dx=-5[/MATH]
Can you proceed?