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Thread: Calculus: finding the intergal of sqrt(cos (6x)+1)

  1. #1

    Calculus: finding the intergal of sqrt(cos (6x)+1)

    Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), Iíve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.



    Find the indefinite integral:

    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx[/tex]



    Step (1)

    Apply rule:


    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(ax)\, +\, 1\,}\bigg)\, dx\, \longrightarrow\, \int\, \bigg(\sqrt{\strut 2\,}\, \cos\left(\dfrac{1}{2}ax\right)\bigg)\, dx[/tex]

    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx\, =\, \dfrac{1}{3}\, \sqrt{\strut 2\,}\, \sin(3x)[/tex]
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    Last edited by stapel; 03-09-2018 at 03:06 PM. Reason: Typing out the text in the graphic; creating useful subject line.

  2. #2

    Calculus

    Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), Iíve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.

  3. #3
    Elite Member
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    Quote Originally Posted by kgilyot View Post
    Hello, I would like to know if you can help me with this problem the intergal of sqrt(cos (6x)+1), Iíve understood it as far as cos (1/2 (ax)) but where is the sqrt of 2 coming from?! Thank you.



    Find the indefinite integral:

    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx[/tex]



    Step (1)

    Apply rule:


    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(ax)\, +\, 1\,}\bigg)\, dx\, \longrightarrow\, \int\, \bigg(\sqrt{\strut 2\,}\, \cos\left(\dfrac{1}{2}ax\right)\bigg)\, dx[/tex]

    . . . . .[tex]\displaystyle \int\, \bigg(\sqrt{\strut \cos(6x)\, +\, 1\,}\bigg)\, dx\, =\, \dfrac{1}{3}\, \sqrt{\strut 2\,}\, \sin(3x)[/tex]
    Hint:

    cos(2Θ) + 1 = 2*cos2(Θ)

    Your attachment is too fuzzy to read.
    Last edited by stapel; 03-09-2018 at 03:06 PM. Reason: Copying typed-out graphical content into reply.
    ď... mathematics is only the art of saying the same thing in different wordsĒ - B. Russell

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