Results 1 to 5 of 5

Math Help - Int/diff of the form f(x+a) dx

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    69

    Int/diff of the form f(x+a) dx

    Can anyone tell me the rules concerning integrating or differentiating functions like the following with respect to x

    \int\cos(a-x) dx

    I have only come across those involving x or multiples of x

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    See…

    http://en.wikipedia.org/wiki/Trigonometric_identity

    ‘… angle sum and difference identities, also known as the addition and subtraction theorems or formulæ. They were originally established by the 10th century Persian mathematician Abū al-Wafā' al-Būzjānī… ‘

    Very interesting! …

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    Quote Originally Posted by bobred View Post
    Can anyone tell me the rules concerning integrating or differentiating functions like the following with respect to x

    \int\cos(a-x) dx

    I have only come across those involving x or multiples of x

    Thanks
    You could use the sum formulas chisigma mentions: cos(a- x)= cos(a)cos(x)+ sin(a)sin(x). That gives (cos(a-x))'= -cos(a)sin(x)+ sin(a)cos(x) and \int cos(a- x)dx= cos(a)sin(x)- sin(a)cos(x)+ C.

    Or you could use the chain rule, for differentiation and a simple substitution of integration ("substitution" is essentially the reverse of the chain rule.). Let u= a- x. Then, by the chain rule \frac{d cos(a-x)}{dx}= \frac{d cos(u)}{dx} = \frac{d cos(u)}{du}\frac{du}{dx}= -sin(u)(-1)= sin(u)= sin(a- x).

    Or, to integrate \int cos(a- x)dx, let u= a- x. Then du= -dx so \int sin(a- x)dx= -\int cos(u)du= -sin(u)+ C= -sin(a-x)+ C.


    Concerned that the sum formula gives -cos(a)sin(x)+ sin(a)cos(x) for the derivative while the chain rule gives sin(a-x)? Don't be. Use the sum formula for sine again: sin(a-x)= sin(a)cos(-x)+ cos(a)sin(-x)= sin(a)cos(x)- cos(a)sin(x). They are exactly the same thing.

    Similarly, -sin(a-x)+ C= -sin(a)cos(-x)- cos(a)sin(-x)+ C= -sin(a)cos(x)+ cos(a)sin(x)+ C. Again, the two methods give the same answer.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2008
    Posts
    1,034
    Thanks
    49
    Just in case a picture helps...

    (the chain rule only, not the trigs). Differentiating...



    ... and integrating...



    As usual, straight continuous lines differentiate downwards, integrate up, the straight dashed line similarly but with respect to the dashed balloon expression.

    Don't integrate - balloontegrate!


    http://www.ballooncalculus.org/forum/top.php
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jan 2009
    Posts
    69
    ... use substitution.

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: May 11th 2010, 11:39 AM
  2. Replies: 1
    Last Post: February 16th 2010, 07:21 AM
  3. Replies: 1
    Last Post: November 27th 2009, 05:33 PM
  4. Replies: 14
    Last Post: May 30th 2008, 06:10 AM
  5. rewrite in slope-intercepr form and general form
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: August 10th 2005, 08:50 PM

Search Tags


/mathhelpforum @mathhelpforum