Results 1 to 6 of 6

Math Help - intergrating cos(x/3) ?

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    12

    intergrating cos(x/3) ?

    I wanted to know why cos (x/3) intergrates to 3sin (x/3) instead of just sin (x/3)??
    Thank you!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    May 2011
    Posts
    54

    Re: intergrating cos(x/3) ?

    use the chain rule
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: intergrating cos(x/3) ?

    Quote Originally Posted by katiethegreat View Post
    I wanted to know why cos (x/3) intergrates to 3sin (x/3) instead of just sin (x/3)??
    Thank you!!
    For solving your integral substitute x/3=u, dx/3=du.


    \int\cos(\frac{x}{3}) \ dx


    \frac{x}{3}=u

    \frac{dx}{3}=du


    \int\cos(\frac{x}{3}) \ dx = \int\cos(u) \ (3du)=3\int\cos(u) \ du=3sin(u)+C

    Returning to x, by putting u=\frac{x}{3}, and we get:


    \int\cos(\frac{x}{3}) \ dx=3sin(\frac{x}{3})+C
    Last edited by mr fantastic; June 16th 2011 at 04:42 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,779
    Thanks
    1519

    Re: intergrating cos(x/3) ?

    When boromir said "use the chain rule", he was referring to the fact that integration is the reverse of differentiating. Differentiating sin(x/3), using the chain rule, you get cos(x/3) times the derivative of x/3, which is 1/3. That is, the derivative of sin(x/3) would be (1/3)cos(x/3). To get just cos(x/3) as the derivative, you need to multiply by 3 to cancel that 1/3: the derivative of 3sin(x/3) is 3(cos(x/3))(1/3)= cos(x/3).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5

    Re: intergrating cos(x/3) ?

    Quote Originally Posted by katiethegreat View Post
    I wanted to know why cos (x/3) intergrates to 3sin (x/3) instead of just sin (x/3)??
    Thank you!!
    When you differentiate 3sin (x/3) and sin (x/3) (and I assume that you know how to correctly do this), which one gives you cos(x/3)?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,397
    Thanks
    760

    Re: intergrating cos(x/3) ?

    the chain rule says: (fog)'(x) = f'(g(x))(g'(x)).

    so if you are going to integrate something of the form f'(g(x)), you need the g'(x) factor in with the integrand (the thing you're integrating).

    in the case of integrating cos(x/3), "f" is the sine funtion, "g" is the function g(x) = x/3. so we need the factor g'(x) = 1/3 under the integral to

    make it work. the only way to do this, is to write cos(x/3) = 3[cos(x/3)](1/3), and take the 3 outside the integral.

    then [cos(x/3)](1/3) is the desired form f'(g(x))(g'(x)) = (fog)'(x), so integrating THAT will give us (fog)(x), that is sin(x/3).

    since the 3 outside the integral is just "along for the ride", we multiply it afterwards. in symbols:

    \int \cos(x/3) dx = \int 3(\cos(x/3))(1/3) dx = 3\int (cos(x/3))(1/3) dx = 3\int (sin(x/3))' dx

     = 3sin(x/3) + C
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. intergrating
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 22nd 2009, 05:29 PM
  2. help intergrating
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 20th 2009, 02:08 AM
  3. Intergrating sin^2 x cos^2 x
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: April 9th 2008, 06:40 AM
  4. Intergrating
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 20th 2007, 06:27 PM
  5. Intergrating
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 15th 2007, 08:34 AM

Search Tags


/mathhelpforum @mathhelpforum