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Integration by parts

  1. Jun 24, 2016 #1
    • Member warned that the homework template is required
    |3^xlog3dx


    I don't even know where to start.
    I know that the formula is
    |u.dv = uv - |v.du
    u=3^x v=log3
     

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  3. Jun 24, 2016 #2
    You don't need by parts to do this. You can use a simple substitution.
    [tex]\int 3^x ln 3 dx[/tex]
    Let [itex]u = 3^x[/itex] , then [itex] du = 3^x ln 3 dx[/itex]
    The integral becomes [tex]\int du [/tex] Integrate and substitute u back.
     
  4. Jun 24, 2016 #3

    BiGyElLoWhAt

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    Log 3 is a constant, which means that dv = 0.
     
  5. Jun 24, 2016 #4
    You don't need integration by parts to find this, its pretty simple, do you know what is the derivative of ##3^x##?
     
  6. Jun 24, 2016 #5
    1. The equation is 3^xlog3
     
  7. Jun 24, 2016 #6
    That's what I wrote, or do you mean to say it's base 10?
     
  8. Jun 24, 2016 #7
    it is base 10, but will it make any difference if it is base 10 or any other bases?
     
  9. Jun 24, 2016 #8

    vela

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    Are you sure? It's pretty common in calculus classes to denote the natural logarithm by log.
     
  10. Jun 24, 2016 #9

    fresh_42

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    ##c = \log_{10} 3## would only be a change by a constant factor ##c##.
    You can write ##3^x \cdot \log_{10} 3 = e^{x \cdot \ln3} \cdot \log_{10}3## and use ##\int c e^{ax} dx = \frac{c}{a} e^{ax} + const.## with ##a = \ln 3.##
     
  11. Jun 24, 2016 #10
    It will make a slight difference, using the same substitution the integral would be [itex]\log e [/itex] (base 10) which is also easily integrated as it's a constant.

    But, like @vela said you should make sure it's base 10.
     
  12. Jun 25, 2016 #11
    ∫3×ln3.dx
    u=3× dx=du/ln(3).3×
    ∫u.ln(3).du/ln(3).3×
    ln(3) cancelled each other.
    ∫u.du/3×
    ⅓×∫u.du
    ⅓×.u²/2
    Then I would substitute u=3×
    Am I right with this?
     
  13. Jun 25, 2016 #12

    SammyS

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    That integrand is 3 times ln(3) .

    I think you mean for it to be 3x ln(3) .

    After the u substitution you have that du = 3x ln(3) dx .

    Thus your integral simply becomes ##\ \int du \ .##
     
  14. Jun 26, 2016 #13
    Okay I got it now, u° equal 1, so the answer is 3×.
     
  15. Jun 26, 2016 #14

    SammyS

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    I believe that you mean 3x or write 3^x .

    Also, don't forget the constant of integration.
     
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