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calculus problems please help?
Let f be a function with a continuous second derivative, f(0) = 10, f(1) = 9, and f '(1) = 9. Evaluate the integral below.
int from 0 to 1 x f ''(x) dx
1 Answer
- gôhpihánLv 71 decade agoFavorite Answer
∫ x f''(x) dx
Integrate by parts
u = x ==> du = dx
dv = f''(x) dx ==> v = f'(x)
∫ x f''(x) dx
= ∫ u dv
= uv - ∫ v du
= x f'(x) - ∫ f'(x) dx
= x f'(x) - f(x) + C
∫(from 0 to 1) x f''(x) dx
= [ x f'(x) - f(x) ] (from 0 to 1)
= [ (1) f'(1) - f(1) ] - [ (0) f'(0) - f(0) ]
= f'(1) - f(1) + f(0)
= 9 - 9 + 10
= 10
