Integrate 1 + sinx/cos x

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  1. This Honest's Avatar
    1 01-04-2012  17:51
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    Integrate 1 + sinx/cos x
    I'm quite stuck on this question: Integrate 1 + sinx/cosx by substitution.
    The book also says let u=sin x

    Now I was thinking of changing it into sec x + tan x but that didn't get me far

    Should I bring cosx to the top?

    Sorry btw, I can't use latex so please bear with me!
  2. nuodai's Avatar
    2 01-04-2012  17:53
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    Re: Integrate 1 + sinx/cos x
    If you substitute u=\sin x then you get dx = \dfrac{du}{\cos x}, so you get \cos^2 x on the denominator. How might you write \cos^2 x in terms of u?
    Last edited by nuodai; 01-04-2012 at 18:36.
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  3. This Honest's Avatar
    3 01-04-2012  17:57
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    Re: Integrate 1 + sinx/cos x
    (Original post by nuodai)
    If you substitute u=\cos x then you get dx = \dfrac{du}{\cos x}, so you get \cos^2 x on the denominator. How might you write \cos^2 x in terms of u?
    U^2

    e: i got negged coz i got it wrong :facepalm2:
    Last edited by This Honest; 01-04-2012 at 18:40.
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  4. This Honest's Avatar
    4 01-04-2012  18:00
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    Re: Integrate 1 + sinx/cos x
    So I got 1 + sinx du/cos^2x?
  5. nuodai's Avatar
    5 01-04-2012  18:01
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    U^2
    No. u=\sin x so u^2 = \sin^2 x \ne \cos^2 x - try again!

    (Original post by This Honest)
    So I got 1 + sinx du/cos^2x?
    Not quite, you should have \dfrac{1 + \sin x}{\cos^2 x} du. Now put this in terms of u.
  6. f1mad's Avatar
    6 01-04-2012  18:01
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    So I got 1 + sinx du/cos^2x?
    Then use the fact that u= sinx

    and cos^2x = 1-sin^2x.
  7. This Honest's Avatar
    7 01-04-2012  18:04
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    Re: Integrate 1 + sinx/cos x
    (Original post by nuodai)
    No. u=\sin x so u^2 = \sin^2 x \ne \cos^2 x - try again!



    Not quite, you should have \dfrac{1 + \sin x}{\cos^2 x} du. Now put this in terms of u.

    (Original post by f1mad)
    Then use the fact that u= sinx

    and cos^2x = 1-sin^2x.
    Alright guys,

    I've now got: 1 + u/1-sin^x
    Last edited by This Honest; 01-04-2012 at 18:06.
  8. nuodai's Avatar
    8 01-04-2012  18:06
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    Alright guys,

    I've now got: 1 + u/1-sin^x
    Write it in terms of u; that still has an x in it. You'll notice that you can factorize something that simplifies the expression into something you can integrate.
  9. raheem94's Avatar
    9 01-04-2012  18:06
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    I'm quite stuck on this question: Integrate 1 + sinx/cosx by substitution.
    The book also says let u=sin x

    Now I was thinking of changing it into sec x + tan x but that didn't get me far

    Should I bring cosx to the top?

    Sorry btw, I can't use latex so please bear with me!
    This question can also be easily done without substitution,

    \displaystyle \int \left(1+\frac{sinx}{cosx}\right) dx

    Now try to differentiate, \displaystyle u=ln(cosx)
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  10. This Honest's Avatar
    10 01-04-2012  18:06
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    Re: Integrate 1 + sinx/cos x
    integrate: 1 +u/1-u^2 ??
  11. blacklistmember's Avatar
    11 01-04-2012  18:07
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    I'm quite stuck on this question: Integrate 1 + sinx/cosx by substitution.
    The book also says let u=sin x

    Now I was thinking of changing it into sec x + tan x but that didn't get me far

    Should I bring cosx to the top?

    Sorry btw, I can't use latex so please bear with me!
    I'm sorry but i'm too busy to help you
    Post rating:
    2
  12. James A's Avatar
    12 01-04-2012  18:08
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    Alright guys,

    I've now got: 1 + u/1-sin^x
    1+u/1-u^2

    remember that u = sinx and that cos^2(x) = 1 - sin^2(x)
  13. This Honest's Avatar
    13 01-04-2012  18:09
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    Re: Integrate 1 + sinx/cos x
    (Original post by nuodai)
    Write it in terms of u; that still has an x in it. You'll notice that you can factorize something that simplifies the expression into something you can integrate.
    :woo: 1/1-u
  14. raheem94's Avatar
    14 01-04-2012  18:09
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    integrate: 1 +u/1-u^2 ??
    Yes, remember you will have to use partial fraction.

    \displaystyle 1 + \frac{u}{1-u^2} = 1 + \frac{u}{(1-u)(1+u)}
  15. This Honest's Avatar
    15 01-04-2012  18:10
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    Re: Integrate 1 + sinx/cos x
    I GOT IT!

    -ln!1-sinx! + c

    can't find modular signs on keyboard $

    Thanks everyone for the help
  16. James A's Avatar
    16 01-04-2012  18:11
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    Re: Integrate 1 + sinx/cos x
    (Original post by This Honest)
    integrate: 1 +u/1-u^2 ??
    have you learnt the integration rule, where if you differentiate the denominator, you get the value that appears in the numerator. hence ln(your original function)

    have you come across this before?

    this could prove useful here.

    someone else correct me if im wrong

    WHOOPS PARTIAL FRACTIONS ARE THE WAY FORWARD :woo:
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  17. This Honest's Avatar
    17 01-04-2012  18:12
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    Re: Integrate 1 + sinx/cos x
    (Original post by raheem94)
    Yes, remember you will have to use partial fraction.

    \displaystyle 1 + \frac{u}{1-u^2} = 1 + \frac{u}{(1-u)(1+u)}
    I didn't quite understand your method but thanks anyways
  18. f1mad's Avatar
    18 01-04-2012  18:12
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    Re: Integrate 1 + sinx/cos x
    (Original post by blacklistmember)
    I'm sorry but i'm too busy to help you
    Why post then?
  19. This Honest's Avatar
    19 01-04-2012  18:13
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    Re: Integrate 1 + sinx/cos x
    (Original post by James A)
    have you learnt the integration rule, where if you differentiate the denominator, you get the value that appears in the numerator. hence ln(your original function)

    have you come across this before?

    this could prove useful here.

    someone else correct me if im wrong

    WHOOPS PARTIAL FRACTIONS ARE THE WAY FORWARD :woo:
    OHHHHHHH YEAHHHHHHH!

    The book said "by sub" so my mind was only on sub and nothing else
  20. f1mad's Avatar
    20 01-04-2012  18:13
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    Re: Integrate 1 + sinx/cos x
    (Original post by raheem94)
    Yes, remember you will have to use partial fraction.

    \displaystyle 1 + \frac{u}{1-u^2} = 1 + \frac{u}{(1-u)(1+u)}
    I don't think so.

    If it's (1+u)/ (1-u^2) = (1+u)/(1-u)(1+u) = 1/(1-u) = - I of -1/(1-u) then it's a simple f'(u)/f(u) integral.

    I think you mis-interpreted .
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