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Integration By Parts help

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data
    1.[tex]$\int x^ne^xdx$[/tex]
    2.[tex]$\int \sin ^nxdx$[/tex]


    2. Relevant equations
    [tex]$ \displaystyle \Large \int fg dx = fg - \int gf' dx$ [/tex]


    3. The attempt at a solution
    1. f=xn
    g'=ex
    g=ex
    f'=nxn-1

    then just plug it in the formula? i tried but i dont get the right answer..

    2. i have no idea how to even start..
    the antiderivative of sinnx is [(sinx)n+1]/(n+1) ?
     
  2. jcsd
  3. Feb 10, 2010 #2

    HallsofIvy

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    No, the anti-derivative of sinn(x) is NOT sinn+1(x)/(n+1)!

    Integrating [itex]\int x^n e^x dx[/itex] is very easy- but tedious. Let u= xn, dv= ex dx. Then du= n xn-1 dx and v= ex.

    [tex]\int x^n e^x dx= x^ne^x- n\int x^{n-1}e^x dx[/tex]
    That is the same as you started with but the exponent on x is one less. Repeat n-1 more times until the exponent is 0!

    As for (2), how you do that depends on whether n is even or odd. If n is odd, it is easy. If n= 2m+1, then the integral is
    [tex]\int sin^{2m+1}(x) dx= \int (sin^2(x))^m sin(x)dx= \int (1- cos^2(x))^msin(x) dx[/tex]
    and letting u= cos(x) reduces it to
    [tex]-\int (1- u^2)^m du[/tex]

    If n is even, use the trig identity [itex]sin^2(x)= (1/2)(1- cos(2x))[/itex] repeatedly until you have reduced to power 1.
     
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