If f is a function such that f ' (x) = - f(x)

Question

then ∫ x f(x) dx =

Answer 1

Use integration by parts.

 

∫udv = uv - ∫vdu

 

u = x

du = dx

dv = f(x)dx = -f'(x)dx ⇒ v = -∫f'(x)dx = -f(x)

 

∫udv = -xf(x) - ∫-f(x)dx = -xf(x) + ∫f(x)dx

 

= -xf(x) - ∫f'(x)dx = -xf(x) - f(x) + C = -(x + 1)f(x) + C

 

Test the solution.