Results 1 to 6 of 6

Math Help - integral problem

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    1

    integral problem

    I need help finding the integral for:

    Cos(x)*E^x

    I've tried integrating by parts, and I keep going in circles, but maybe there's a trick I'm missing.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,929
    Thanks
    757
    \int e^x \cos{x} \, dx

    u = \cos{x}

    du = -\sin{x} \, dx

    dv = e^x \, dx

    v = e^x

    \int e^x \cos{x} \, dx = e^x \cos{x} + \int e^x \sin{x} \, dx

    use parts again ...

    u = \sin{x}

    du = \cos{x} \, dx

    dv = e^x \, dx

    v = e^x

    \int e^x \cos{x} \, dx = e^x \cos{x} + e^x \sin{x} - \int e^x \cos{x} \, dx

    2\int e^x \cos{x} \, dx = e^x \cos{x} + e^x \sin{x}

    \int e^x \cos{x} \, dx = \frac{e^x \cos{x} + e^x \sin{x}}{2} + C
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by andyourluckynumberis3.14 View Post
    I need help finding the integral for:

    Cos(x)*E^x

    I've tried integrating by parts, and I keep going in circles, but maybe there's a trick I'm missing.
    An alternative approach if you're familiar with complex variable theory:

    Note that \cos (x) \, e^x = \text{Re} \left[e^{ix} \cdot e^x\right] = \text{Re}\left[e^{(1 + i)x}\right].

    Then the integral becomes \int \text{Re} \left[e^{(1 + i)x}\right] \, dx = \text{Re} \int e^{(1 + i)x} \, dx

    (reversing the order of the operators requires justification of course).

    Now do the simple integral and take the real part of the result.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Dec 2008
    From
    Scotland
    Posts
    901
    And yet another approach would be to let:  I = \int e^x cos(x) , and integrate twice using parts. When you do that, you end up generating I on the RHS.

     I = e^xcos(x) - \int e^x (-sin(x)) dx

     I = e^x cos(x) + e^xsin(x) -\int e^xcos(x)dx + C^*

    Which of course gives:

     I = e^x cos(x) + e^xsin(x) - I +C^*

    And hence can solve algebraically for I to get:

     I = \frac{ e^x cos(x) + e^xsin(x)}{2} + C

    (where C = \frac{C^*}{2})
    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
    Quote Originally Posted by Mush View Post
    And yet another approach would be to let:  I = \int e^x cos(x) , and integrate twice using parts. When you do that, you end up generating I on the RHS.

     I = e^xcos(x) - \int e^x (-sin(x)) dx

     I = e^x cos(x) + e^xsin(x) -\int e^xcos(x)dx + C^*

    Which of course gives:

     I = e^x cos(x) + e^xsin(x) - I +C^*

    And hence can solve algebraically for I to get:

     I = \frac{ e^x cos(x) + e^xsin(x)}{2} + C

    (where C = \frac{C^*}{2})
    Echoes of post #2 ....?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Dec 2008
    From
    Scotland
    Posts
    901
    Quote Originally Posted by mr fantastic View Post
    Echoes of post #2 ....?
    Ah very sorry, didn't see that! I just read the first few lines of that post with all the u v stuff and for some reason assumed he was using some variation of integration by substitution, didn't realise is was equivalent to my method.

    Apologies.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. integral problem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 12th 2011, 09:51 PM
  2. Integral problem involving the definition of the integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 23rd 2011, 07:06 AM
  3. Integral Problem Help
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 21st 2010, 07:35 AM
  4. Integral Problem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: September 25th 2009, 06:11 AM
  5. integral problem...
    Posted in the Calculus Forum
    Replies: 16
    Last Post: April 24th 2009, 08:58 AM

Search Tags


/mathhelpforum @mathhelpforum