Page 1 of 2 12 LastLast
Results 1 to 25 of 27
Like Tree8Likes

Thread: Integration

  1. #1
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Integration

    How do you integrate xdx/sqrt(x-x^2) (exercise 4.3 question 1h) from terry lee)?

    Thanks for the help
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  2. #2
    Loquacious One
    Join Date
    Feb 2009
    HSC
    N/A
    Gender
    Male
    Posts
    3,652
    Rep Power
    9

    Re: Integration

    Last edited by Drongoski; 14 Feb 2017 at 2:25 PM.
    pikachu975 likes this.
    1-on-1 Maths Tutoring(IB & HSC): Epping, Beecroft, Eastwood, Carlingford & Beyond
    IB: Maths Studies, Maths SL & Maths HL; HSC: 2U, 3U & 4U
    Highly Qualified & Highly Experienced. Estimated ATAR > 9.995
    There are IB Maths Tutors and there are IB Maths Tutors.

  3. #3
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by Drongoski View Post
    Thanks for the help. Do you mind explaining your thought process for getting the first line where you knew to do -0.5 (1-2x) + 0.5? Thanks again
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  4. #4
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    Hey I know it's been answered but I still wanna upload it so I can ask a question. pikachu integral image.png. How can I put the latex in the actual post like the others above? I only know how to link as a picture
    1st Year BAppFinBActStud @ MQ

  5. #5
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by BenHowe View Post
    Hey I know it's been answered but I still wanna upload it so I can ask a question. pikachu integral image.png. How can I put the latex in the actual post like the others above? I only know how to link as a picture
    Did you type it as a code? Click reply to Drongoski's post and see the code required for latex
    BenHowe likes this.
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  6. #6
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    Ok lemme try [tex] $\int\frac{x}{\sqrt{x-x^{2}}}dx$

    $=-\frac{1}{2}\int\frac{-2x}{\sqrt{-x^{2}+x}}dx$

    $=-\frac{1}{2}\int\frac{-2x+1}{\sqrt{-x^{2}+x}}-\frac{1}{\sqrt{-x^{2}+x}}dx$

    $ now -x^{2}+x=-(x^{2}-x)$

    $=-((x-\frac{1}{2})^{2}-\frac{1}{4})$

    $=(\frac{1}{2})^{2}-(x-\frac{1}{2})^{2}$

    $=-\frac{1}{2}\int\frac{-2x+1}{\sqrt{-x^{2}+x}}-\frac{1}{\sqrt{(\frac{1}{2})^{2}-(x-\frac{1}{2})^{2}}}dx$

    $=-\sqrt{-x^{2}+x}+\frac{1}{2}sin^{-1}(2x-1)+c\hspace{0.1cm}$ [\tex]
    1st Year BAppFinBActStud @ MQ

  7. #7
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by BenHowe View Post
    Ok lemme try [tex] $\int\frac{x}{\sqrt{x-x^{2}}}dx$

    $=-\frac{1}{2}\int\frac{-2x}{\sqrt{-x^{2}+x}}dx$

    $=-\frac{1}{2}\int\frac{-2x+1}{\sqrt{-x^{2}+x}}-\frac{1}{\sqrt{-x^{2}+x}}dx$

    $ now -x^{2}+x=-(x^{2}-x)$

    $=-((x-\frac{1}{2})^{2}-\frac{1}{4})$

    $=(\frac{1}{2})^{2}-(x-\frac{1}{2})^{2}$

    $=-\frac{1}{2}\int\frac{-2x+1}{\sqrt{-x^{2}+x}}-\frac{1}{\sqrt{(\frac{1}{2})^{2}-(x-\frac{1}{2})^{2}}}dx$

    $=-\sqrt{-x^{2}+x}+\frac{1}{2}sin^{-1}(2x-1)+c\hspace{0.1cm}$ [\tex]
    You don't press enter for a new line maybe it is \\
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  8. #8
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    pikachu975 likes this.
    1st Year BAppFinBActStud @ MQ

  9. #9
    Loquacious One
    Join Date
    Feb 2009
    HSC
    N/A
    Gender
    Male
    Posts
    3,652
    Rep Power
    9

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    Thanks for the help. Do you mind explaining your thought process for getting the first line where you knew to do -0.5 (1-2x) + 0.5? Thanks again
    Using my personal way of integrating, I note that the derivative of x-x2 is 1-2x. so I prepare the denominator of the integrand to have a multiple of 1-2x with some remainder. What I have done is to fiddle the integrand into 2 parts that are easily integrable. Because I have been doing this for many many many years, I can see what I need to do.
    1-on-1 Maths Tutoring(IB & HSC): Epping, Beecroft, Eastwood, Carlingford & Beyond
    IB: Maths Studies, Maths SL & Maths HL; HSC: 2U, 3U & 4U
    Highly Qualified & Highly Experienced. Estimated ATAR > 9.995
    There are IB Maths Tutors and there are IB Maths Tutors.

  10. #10
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by Drongoski View Post
    Using my personal way of integrating, I note that the derivative of x-x2 is 1-2x. so I prepare the denominator of the integrand to have a multiple of 1-2x with some remainder. What I have done is to fiddle the integrand into 2 parts that are easily integrable. Because I have been doing this for many many many years, I can see what I need to do.
    How would you integrate x^2 dx / (x^2 - 2x + 3) lol sorry integration is so hard....

    I got x + ln|x^2 - 2x+3| - (tan^-1 (x-1 / root2))/root 2 but it looks weird
    Last edited by pikachu975; 14 Feb 2017 at 2:54 PM.
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  11. #11
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    Thanks for the help. Do you mind explaining your thought process for getting the first line where you knew to do -0.5 (1-2x) + 0.5? Thanks again






    pikachu975 likes this.

  12. #12
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    How would you integrate x^2 dx / (x^2 - 2x + 3) lol sorry integration is so hard....

    I got x + ln|x^2 - 2x+3| - (tan^-1 (x-1 / root2))/root 2 but it looks weird
    pikachu975 likes this.

  13. #13
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by InteGrand View Post






    Thanks a lot for this. How about with a higher degree on the numerator - e.g. x^3 dx / (x^2 + x + 1)
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  14. #14
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    Thanks a lot for this. How about with a higher degree on the numerator - e.g. x^3 dx / (x^2 + x + 1)


    Note that in your previous one where I added and subtracted something to get the numerator of lower degree than the denominator was actually just a shortcut method of polynomial division. (So even if the numerator has equal degree to the denominator, we need to divide first so that the numerator degree becomes strictly less than the denominator degree, after which we can use the earlier outlined method.)

    Also note that if the denominator is a quadratic (without square root) with two easy roots and the numerator is a linear function (including constant function), we can also use the method of partial fractions relatively easily.
    Last edited by InteGrand; 14 Feb 2017 at 5:14 PM.

  15. #15
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    How would you integrate x^2 dx / (x^2 - 2x + 3) lol sorry integration is so hard....

    I got x + ln|x^2 - 2x+3| - (tan^-1 (x-1 / root2))/root 2 but it looks weird
    I think you thought your question was not cause of the sqrt on the denominator in the 1st q

    Makes a substantial difference
    Last edited by BenHowe; 14 Feb 2017 at 3:46 PM.
    1st Year BAppFinBActStud @ MQ

  16. #16
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by BenHowe View Post
    I think you thought your question was not cause of the sqrt on the denominator in the 1st q

    Makes a substantial difference
    I did what integrand said and added and subtracted something from the numerator

    Quote Originally Posted by InteGrand View Post


    Note than in your previous one where I added and subtracted something to get the numerator of lower degree than the denominator was actually just a shortcut method of polynomial division. (So even if the numerator has equal degree to the denominator, we need to divide first so that the numerator degree becomes strictly less than the denominator degree, after which we can use the earlier outlined method.)

    Also note that if the denominator is a quadratic (without square root) with two easy roots and the numerator is a linear function (including constant function), we can also use the method of partial fractions relatively easily.
    Thanks for the response but why do we use the remainder?
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  17. #17
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    I did what integrand said and added and subtracted something from the numerator



    Thanks for the response but why do we use the remainder?
    Its degree is less than the denominator's so we can use the method outlined earlier for that part.

  18. #18
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by InteGrand View Post
    Its degree is less than the denominator's so we can use the method outlined earlier for that part.
    So for example the remainder is x, then you use x/(x^2+x+1) ? Why does it still give the same answer?
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  19. #19
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by pikachu975 View Post
    So for example the remainder is x, then you use x/(x^2+x+1) ? Why does it still give the same answer?
    pikachu975 likes this.

  20. #20
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration



    Last edited by InteGrand; 14 Feb 2017 at 4:57 PM.

  21. #21
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by InteGrand View Post


    Thank you for the help as well as Drongoski and BenHowe
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

  22. #22
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    Quote Originally Posted by InteGrand View Post


    How did you write the quotient and remainder part underneath the terms, it looks so epic
    pikachu975 likes this.
    1st Year BAppFinBActStud @ MQ

  23. #23
    Junior Member BenHowe's Avatar
    Join Date
    Aug 2015
    HSC
    2016
    Gender
    Male
    Location
    Sydney
    Posts
    234
    Rep Power
    2

    Re: Integration

    Pikachu the only reason I said that you mighta thought it had the square root in the denominator because originally I thought it did as well and got the same answer as you
    1st Year BAppFinBActStud @ MQ

  24. #24
    Ancient Orator
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    5,970
    Rep Power
    6

    Re: Integration

    Quote Originally Posted by BenHowe View Post
    How did you write the quotient and remainder part underneath the terms, it looks so epic
    I used

    Code:
    \underbrace{...}_{\text{...}}
    .

    (Replace the first "..." with the thing you want the brace to go under and the second "..." with what text you want to write under the brace.)
    BenHowe likes this.

  25. #25
    I love trials pikachu975's Avatar
    Join Date
    May 2015
    HSC
    2017
    Gender
    Male
    Location
    NSW
    Posts
    2,025
    Rep Power
    3

    Re: Integration

    Quote Originally Posted by BenHowe View Post
    Pikachu the only reason I said that you mighta thought it had the square root in the denominator because originally I thought it did as well and got the same answer as you
    I checked I didn't have it, I used integrand's method stated somewhere above
    Offering tutoring for Biology, Maths, Maths Ext 1, and Maths Ext 2.

    Solar Education - CHEAP Quality Tutoring

    Buy my books/notes cheaply here!

    --------------------------------------------------------------------------------

    2016 HSC (Accelerated): // 2U Maths (97) // SOR 1 (48) //
    2017 HSC: // English Adv // Bio // Phys // 3U Maths // 4U Maths //
    Goal: 99.8+

Page 1 of 2 12 LastLast

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •