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Integrate a function

  1. Jan 26, 2014 #1
    1. The problem statement, all variables and given/known data

    This function of one variable need to be integrated: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi, without help with calculators or computer algebra software.


    2. Relevant equations

    Have no idea.

    3. The attempt at a solution

    Have no idea.

    I don't have any idea how to start. Please help with a hint :).
     
    Last edited: Jan 26, 2014
  2. jcsd
  3. Jan 26, 2014 #2

    Ray Vickson

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    Do you have a textbook or course notes? If so, start there.
     
  4. Jan 26, 2014 #3
    Nothing found in the textbook which would be useful.
     
  5. Jan 27, 2014 #4

    Office_Shredder

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    It is unlikely that you have been assigned a homework question which has nothing to do with the material that has been taught in your class or written in your textbook, if you cannot find a single example of a similar looking integral being solved then it probably means you need to look again.
     
  6. Jan 27, 2014 #5
    Hi there,

    How would one integrate a function like this: f(x)=1/(9+7*(sinx)^2) from x=0 to x=pi?

    I have no idea how to start. Any hint or help will be highly appreciated.
     
  7. Jan 27, 2014 #6

    Attached Files:

  8. Jan 28, 2014 #7
    What should that imply?

    Anyway, I want to integrate the function by pencil and paper, not by computer algebra software, calculator or any electronic device. I will be very thankful about any hint.
     
  9. Jan 28, 2014 #8

    SteamKing

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    Take a look at a table of integrals and see if your function is similar to one of the integrals in the table.

    Just eyeballing what you have, either a trigonometric or hyperbolic substitution would be used here.
     
  10. Jan 28, 2014 #9
    Divide denominator and numerator by cos^2(x). Then substitute tan^2(x) in the denominator as T (or any other variable you want). Then proceed with the procedure for solving integration by substitution.
    If you have problem in solving this question, let me know.
     
  11. Jan 28, 2014 #10

    berkeman

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    (Moderator's Note -- 2 threads merged)
     
  12. Jan 28, 2014 #11

    Dick

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    Try writing 9+7*sin(x)^2 as 9*(sin(x)^2+cos(x)^2)+7*sin(x)^2. That might give you an idea.
     
  13. Jan 28, 2014 #12

    Ray Vickson

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    Since your textbook is apparently useless, go on-line: a Google search under "integration methods" produces a host of useful websites, including http://www.mathwords.com/i/integration_methods.htm or http://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx or http://math.colorado.edu/~prestos/math2300/math2300integral.pdf [Broken] . This last one seems particularly relevant to your issues.
     
    Last edited by a moderator: May 6, 2017
  14. Jan 29, 2014 #13
    You are 28, not 18, so...

    I meant: you are not a young student looking for solutions to a homework; so. I tought you were a technician or a researcher needing that integral for some computations. In such cases, the fast way is just Wolfram Alpha.

    Reading also what you have written in the merged thread, now I understand you are not looking for a solution for that particular integral, but rather for an explanation of what "integral" means...

    Well, if this is the case, to solve

    [itex]\int[/itex] 0 π f(x) dx

    with

    f(x)=1/(9+7*(sin x)2)

    you have to:

    1) find a primitive, AKA indefinite integral, AKA antiderivative (http://en.wikipedia.org/wiki/Antiderivative) of f(x), i.e. a function F(x) whose derivative is just f(x).

    Of course, if F(x) is such a function, also F(x)+k (with any number in place of k) is a primitive of f(x), given that the derivative of a constant is zero.


    2) compute F(π) - F(0). Of course, with any k, F(π) + k - [F(0) + k] will give the same numerical result.

    The latter point, 2), is due to the Fundamental theorem of calculus http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus. It's demonstration is rather intuitive, have a look.

    For what concerns point 1), while finding the derivative of a function can be tedious but is straightforward using simple rules, finding a primitive is often considerably hard. There are some techniques and some tricks that can be attempted, hoping that they could lead to the solution: but knowing in advance which could be more promising is a matter of experience (acquired after several homeworks :- ). Other post in this thread have suggested you some ways.

    On the contrary, once somebody or some software has found the primitive you need (Wolfram Alpha gave it in the section "indefinite integral"), checking that it is the right solutions requires only to derive it.

    Zdravo
     
    Last edited: Jan 29, 2014
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