Results 1 to 6 of 6

Math Help - integrate cos^3(x)

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    68

    integrate cos^3(x)

    Hi, how do I integrate cos^3(x)?

    I split it into cos(x)[1/2(1+cos(2x)] and then removed the constant of a 1/2 from the integration. Then I mutilpied through by cos(x) and got cos(x)+cos^2(2x), which didn't really help.

    Then I split cos^3(x) into cos(x).cos^2(x) tried substituting cos^2(x) with 1-sin^2(x) and that didn't help either.

    What am I missing?
    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by slaypullingcat View Post
    Hi, how do I integrate cos^3(x)?

    I split it into cos(x)[1/2(1+cos(2x)] and then removed the constant of a 1/2 from the integration. Then I mutilpied through by cos(x) and got cos(x)+cos^2(2x), which didn't really help.

    Then I split cos^3(x) into cos(x).cos^2(x) tried substituting cos^2(x) with 1-sin^2(x) and that didn't help either.

    What am I missing?
    Thank you.
    Note that \int\cos^3x\,dx=\int\cos^2x\cos x\,dx=\int\left(1-\sin^2x\right)\cos x\,dx =\int\cos x\,dx-\int\sin^2x\cos x\,dx

    In the second integral, make the substitution u=\sin x.

    Can you continue?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2009
    Posts
    68
    Quote Originally Posted by Chris L T521 View Post
    Note that \int\cos^3x\,dx=\int\cos^2x\cos x\,dx=\int\left(1-\sin^2x\right)\cos x\,dx =\int\cos x\,dx-\int\sin^2x\cos x\,dx

    In the second integral, make the substitution u=\sin x.

    Can you continue?
    I will certainly try. Thank you for your post.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,569
    Thanks
    1428
    Quote Originally Posted by slaypullingcat View Post
    Hi, how do I integrate cos^3(x)?

    I split it into cos(x)[1/2(1+cos(2x)] and then removed the constant of a 1/2 from the integration. Then I mutilpied through by cos(x) and got cos(x)+cos^2(2x), which didn't really help.

    Then I split cos^3(x) into cos(x).cos^2(x) tried substituting cos^2(x) with 1-sin^2(x) and that didn't help either.

    What am I missing?
    Thank you.
    Use the identity

    \cos^3{x} = \frac{3}{4}\cos{x} + \frac{1}{4}\cos{(3x)}.


    So

    \int{\cos^3{x}\,dx} = \int{\frac{3}{4}\cos{x} + \frac{1}{4}\cos{(3x)}\,dx}

     = \frac{3}{4}\sin{x} + \frac{1}{12}\sin{(3x)} + C.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jan 2009
    Posts
    68
    Quote Originally Posted by Chris L T521 View Post
    Note that \int\cos^3x\,dx=\int\cos^2x\cos x\,dx=\int\left(1-\sin^2x\right)\cos x\,dx =\int\cos x\,dx-\int\sin^2x\cos x\,dx

    In the second integral, make the substitution u=\sin x.

    Can you continue?
    Out of curiosity, is there a method of solving this problem without using substitution. In the cousre I am doing substitution has already been covered so I am fine with that, but I am trying to get the big picture of integration and get a feel for as many techniques I can. Cheers
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,569
    Thanks
    1428
    Quote Originally Posted by slaypullingcat View Post
    Out of curiosity, is there a method of solving this problem without using substitution. In the cousre I am doing substitution has already been covered so I am fine with that, but I am trying to get the big picture of integration and get a feel for as many techniques I can. Cheers
    Yes, use the identity I've given you. There's no u substitution there...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integrate 3 e^x
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 9th 2010, 01:16 PM
  2. how to integrate this ??
    Posted in the Calculus Forum
    Replies: 8
    Last Post: August 9th 2009, 08:14 PM
  3. Integrate
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 28th 2009, 07:54 PM
  4. How to integrate x^x?
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 25th 2009, 07:51 PM
  5. Integrate
    Posted in the Calculus Forum
    Replies: 8
    Last Post: April 15th 2009, 03:50 PM

Search Tags


/mathhelpforum @mathhelpforum