Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    Im having trouble integrating these 2:

    ∫x2 /x2 +x-20.dx
    and
    ∫(x2 +1/3)/(x3+x-1)dx

    Thanks
    Offline

    12
    ReputationRep:
    (Original post by wibble...)
    1.) ∫x2 /x2 +x-20.dx
    2.)∫(x2 +1/3)/(x3+x-1)dx
    1.) ∫x²/(x² + x - 20) dx
    = ∫x²/[(x + 5)(x - 4)] dx
    = ∫A + B/(x + 5) + C/(x - 4) dx
    = ∫{A(x + 5)(x - 4) + B(x - 4) + C(x + 5)}/[(x + 5)(x - 4)] dx

    Equating numerators:
    => A(x + 5)(x - 4) + B(x - 4) + C(x + 5) = x²
    Let x = 4: => 9C = 16 => C = 16/9
    Let x = -5: => -9B = 25 => B = -25/9
    Let x = 0: -20A - 4B + 5C = 0
    => -20A + 4(25/9) + 5(16/9) = 0
    => -20A + 100/9 + 80/9 = 0
    => -20A = - 180/9 = 20
    => A = -1

    :. ∫x²/(x² + x - 20) dx
    = ∫-1 - 25/[9(x + 5)] + 16/[9(x - 4)] dx
    = (-1/9) ∫9 + 25/(x + 5) - 16/(x - 4) dx
    = (-1/9){9x + 25ln|x + 5| - 16ln|x - 4|} + k

    2.) ∫(x² + 1/3)/(x³ + x - 1) dx
    = (1/3) ∫ (3x² + 1)/(x³ + x - 1) dx
    = (1/3) ln|x³ + x - 1| + k
    Offline

    20
    ReputationRep:
    (Original post by wibble...)
    Im having trouble integrating these 2:

    ∫x2 /(x2 +x-20).dx
    you need to rewrite it in partial fractions before you integrate.

    =>∫5/{9(x + 5)} +∫4/{(x - 4)} which will no doubt require ln....


    ∫(x2 +1/3)/(x3+x-1)dx is easier...the top is 1/3 of the differential of the bottom so you can write the answer straight away as

    1/3ln|x3+x-1| + c
    Offline

    0
    ReputationRep:
    your answer for 1) isn't right you can't use partial fractions if the numerator is of the same order as the denominator
    Offline

    0
    ReputationRep:
    Complete the square on the bottom line, giving

    (x + ½)² - 20.25.

    Then let u = x + 1/2, so you get

    (u-0.5)^2/u^2 - 20.25 du

    = ∫u^2 - u + 0.25/u^2 - 20.25 du

    = (∫u^2/u^2 - 20.25 du) - (∫u/u^2 - 20.25 du ) +0.25( ∫1/u^2 - 20.25 du)

    Problem should be soluble from there using standard techniques
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 16, 2005

University open days

  • University of East Anglia
    UEA Mini Open Day Undergraduate
    Fri, 23 Nov '18
  • Norwich University of the Arts
    Undergraduate Open Days Undergraduate
    Fri, 23 Nov '18
  • Edge Hill University
    All Faculties Undergraduate
    Sat, 24 Nov '18
Poll
Black Friday: Yay or Nay?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.