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Substitution and Integration by Parts

  1. Feb 20, 2014 #1
    1. The problem statement, all variables and given/known data
    First make a substitution and then use integration by parts to evaluate the integral.

    ∫x[itex]^{7}[/itex]cos(x[itex]^{4}[/itex])dx

    2. Relevant equations

    Equation for Substitution: ∫f(g(x))g'(x)dx = ∫f(u)du
    Equation for Integration by Parts: ∫udv = uv - ∫vdu

    3. The attempt at a solution

    So here's my attempted solution
    tumblr_n1aepoItwY1tsd2vco1_500.jpg

    I made a substitution and tried using integration by parts twice but I got stuck on the last line since it turns out to be zero... I know I went wrong somewhere but I can't seem to find my mistake. Any help would be really appreciated! Thanks :)
     
  2. jcsd
  3. Feb 20, 2014 #2

    CAF123

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    Gold Member

    The second line is incorrect - when you made the u substitution you did not use your expression for dx in terms of du.
     
  4. Feb 20, 2014 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Start by writing the integral as [itex]\int x^4cos(x^4)(x^3dx)[/itex] and it is clearer.
     
  5. Feb 20, 2014 #4
    I see my mistake now! Thanks for the help :)
     
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