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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You should use integration by parts such that:

`int udv = uv - int vdu`

Considering `u = x`  and `dv = f''(x)`  yields:

`u = x => du = dx`

`dv = f''(x)dx => v = f'(x)`

`int_1^4 xf''(x)dx = xf'(x)|_1^4- int_1^4 f'(x)dx`

`int_1^4 xf''(x)dx = xf'(x)|_1^4 - f(x)|_1^4`

`int_1^4 xf''(x)dx = 4*f'(4) - 1*f'(1) - f(4) + f(1)`

Notice that the problem provides the following values `f(1) = 2; f(4) = 6; f'(1) = 3`  and `f'(4) = 5`  such that:

`int_1^4 xf''(x)dx = 4*5 - 3 - 6 + 2 `

`int_1^4 xf''(x)dx = 13`

Hence, evaluating the definite integral `int_1^4 xf''(x)dx,`  under the given conditions, yields `int_1^4 xf''(x)dx = 13` .

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial