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Integration of a function with it's derivative.

  1. Jan 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Please help me by pointing me in the right direction for this integral :

    ∫ t^3 * (t^2 + 1)^0.5 dt


    2. Relevant equations

    i used the standard formula of integration by parts on this.

    http://www.sosmath.com/calculus/integration/byparts/img9.gif
    3. The attempt at a solution

    here is what is did :
    1/2∫ 2t * t^2 * (t^2 + 1)^0.5 dt
    1/2{ [t^2 * (t^2 + 1)^1.5 * 2/3] - [∫ 2t * (t^2 + 1)^1.5 * 2/3 dt ]}

    i highly doubt it is correct, i think i used the formula wrongly.

    Can anyone point out my mistake or just provide me with any other sort of help, it would be appreciated.
     
  2. jcsd
  3. Jan 23, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    The title of your thread is "Integration of a function with it's derivative". The integral can be written as

    [tex]\int{t^3(t^2+1)^{0.5} dt}=0.5\int{((t^2+1)-1)(t^2+1)^{0.5} (2tdt)}[/tex] .

    Apply "integration by substitution".

    [tex]\int{f(g(x))g'(x)dx}=\int{f(g)dg}[/tex]

    ehild
     
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