Assume that f is a function with a continuous second derivative f''(x) and suppose that f(1) = 2; f(4) = 6; f'(1) = 3 and f'(4) = 5.  Evaluate integrate from 1 to 4 of xf''(x)dx

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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This can be done by integral by parts.

 

Let;

`U = x`

`V = f'(x)`

 

Then;

`dU = dx`

`dV = f''(x)dx`

 

`int UdV = UV-intVdu`

`intxf''(x)dx = xf'(x)-int f'(x)dx`

 

By applying the limits;

`int^4_1xf''(x)dx = [xf'(x)]^4_1-int^4_1 f'(x)dx`

 

`int^4_1xf''(x)dx`

`= [xf'(x)]^4_1-[f(x)]^4_1`

`= 4f'(4)-f'(1)-(f(4)-f(1))`

`= 4*5-3-(6-2)`

`= 13`

 

`int^4_1xf''(x)dx = 13`

 

 

 

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