1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integration by Parts of x^5cos(x^3)

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\int x^5cos(x^3) dx[/tex]

    2. Relevant equations
    [tex]\int uv' = uv - \int u'v[/tex]


    3. The attempt at a solution
    [tex]\int x^5cos(x^3) dx[/tex]

    [tex]\frac{(x^5)*(sin(x^3)}{(3x^2)} - \int\frac{5x^4*sin(x^3)}{(3x^2)} dx[/tex]

    [tex]\frac{(x^3)*sin(x^3)}{3} - \int\frac{(5x^2)*sin(x^3)}{(3)} dx[/tex]

    [tex]\frac{(x^3)*sin(x^3)}{3} - \frac{5}{3} \int x^2*sin(x^3) dx[/tex]

    [tex]\int x^2*sin(u) *\frac{du}{3x^2}[/tex]

    [tex]\frac{1}{3} \int sin(u) du[/tex]

    [tex]\frac{-1}{3} cos(u)[/tex]

    [tex]\frac{-1}{3} cos(x^3)[/tex]

    Revisting

    [tex]\frac{(x^3)*sin(x^3)}{3} - \frac{5}{3} * \frac{-1}{3} cos(x^3)[/tex]

    My answer:

    [tex]\frac{(x^3)*sin(x^3)}{3} + \frac{5}{9} * cos(x^3) + C[/tex]

    Correct Answer:

    [tex]\frac{(x^3)*sin(x^3)}{3} + \frac{1}{3} * cos(x^3) + C[/tex]
     
  2. jcsd
  3. Apr 30, 2009 #2
    no, your first u-dv doesn't work that way. think compostion chainrule and u-substitution.
     
  4. May 1, 2009 #3

    HallsofIvy

    User Avatar
    Science Advisor

    The integral of [itex]cos(x^3)[/itex] is not [itex]sin(x^3)/3x^2[/itex]!

    First write the integral as
    [tex]\int x^3\left[x^2cos(x^3)\right]dx[/tex]
    and let [itex]u= x^3[/itex].
     
  5. May 2, 2009 #4
    May I ask what the derivative of sin(x^3)/3x^2 may be then?

    Thanks, rewriting the function got me the right answer.
     
    Last edited: May 2, 2009
  6. May 2, 2009 #5

    HallsofIvy

    User Avatar
    Science Advisor

    I don't know why you would ask that. It might be helpful to recognize that, by the chain rule, the derivative of [itex]sin(x^3)[/itex] is [itex](2x^2)cos(x^3)[/itex].
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook