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We're given that f(0) = 1 and f(2) = 5, and that
[tex]\displaystyle \int_0^2 f(x) \, dx = 7[/tex]
Integrate x f'(x) by parts, with
[tex]u = x \implies du = dx[/tex]
[tex]dv = f'(x) \, dx \implies v = f(x)[/tex]
Then
[tex]\displaystyle \int_0^2 x f'(x) \, dx = uv\bigg|_{x=0}^{x=2} - \int_0^2 v \, du \\\\ = x f(x) \bigg|_{x=0}^{x=2} - \int_0^2 f(x) \, dx[/tex]
Now just plug in everything you know:
[tex]\displaystyle \int_0^2 x f'(x) \, dx = 2f(2) - 0f(0) - \int_0^2 f(x) \, dx = 10 - 0 - 7 = \boxed{3}[/tex]