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2008
Sun
12
Oct
Miss Loi avatar

Integration Amid Global Disintegration

(5)
Tuition given in the topic of A-Maths Tuition Notes & Tips from the desk of Miss Loi at 7:31 pm (Singapore time)
Death By Integration

May the The Temple never see this
(click to enlarge)

With less than ten days (at the time of writing) to go, the skies darkened as the wind picked up, tossing helpless NTUC Fairprice plastic bags up and down like a stock market, as old symptoms (and a couple of new ones) began to surface once more.

While the world deals with the small matter of the possible disintegration of its global financial system, students at Miss Loi’s Temple have far more important things to worry about, like that big A-Maths Integration chapter.

Eh Miss Loi, how come these two Integration questions I never see before in my TYS one?

The One That Got Away

Latest Edition of Shing Lee Textbook
that missed its intended target

*Narrowly dodging the big Shing Lee A-Maths textbook that came hurling straight at his face He sat upright and was all ears …

Ooi! Dreaming of your crush again?!

You should know by now that two elements from the New A-Maths Syllabus have impishly made their way into the Integration chapter!

GEN-Y INTEGRATON

A. Integration of 1/(ax + b)

Starting this year, we have a new entry to the Integration scene:

For x > 0,
int{}{}{1/{ax + b}} dx = {1/a} ln (ax + b) + c —– (A1)

Having signed a formidable pact with Partial Fractions, the integrals will generally appear in the form of
int{}{}{1/{(ax+b)(cx+d)}} dx or int{}{}{{kx+h}/{(ax+b)(cx+d)}} dx
i.e. only linear factors/repeated linear factors in the denominator – very unlikely you’re see quadratic factors like (x2+b2) unless it’s a Hence question with a big clue in the first part.

So whenever you encounter something like the above, regardless of whether you’ve been explicitly told by the question to do so, you MUST perform your Partial Fractions 三大招式 (Three Devastating Moves) to express it in partial fractions first before proceeding to integrate using (A1) above.

SAMPLE QUESTION

  1. Express {8x+13}/{(1+2x)(2+x)^2} in partial fractions. Hence evaluate int{1}{2}{{8x+13}/{(1+2x)(2+x)^2}} dx.

    Ans:

    Expressing in partial fractions (reference here for the steps if you’re unsure):

    1. Check: degree of (8x+13) = 1 < degree of (1+2x)(2+x)2 = 3
      PROPER (YAY!)
    2. {8x+13}/{(1+2x)(2+x)^2} = A/{1+2x} + B/{2+x} + C/(2+x)^2
      → 1 x linear factor (1+2x) and
      → 1 x repeated linear factor (2+x)2

    3. Multiply both sides by (1+2x)(2+x)2,
      8x+13 = A(2+x)2 + B(1+2x)(2+x) + C(1+2x)

      Sub x = -2, -½ … blah blah blah … we get

      {8x+13}/{(1+2x)(2+x)^2} = 4/{1+2x} - 2/{2+x} + 1/(2+x)^2

      Attention For all rogue Cover-Up Rule users, note that you can only use it to find the corresponding partial fractions for the terms with linear factors and the higher order of repeated linear factors

      e.g. For this question you can use the Cover-Up rule to find A/{1+2x} and C/(2+x)^2 but NOT B/{2+x}.

    So,
    int{1}{2}{{8x+13}/{(1+2x)(2+x)^2}} dx = int{1}{2}{4/{1+2x} - 2/{2+x} + 1/(2+x)^2} dx
    {} = 4 int{1}{2}{1/{1+2x}} dx - 2 int{1}{2}{1/{2+x}} dx + int{1}{2}{1/(2+x)^2} dx

    And now you can unleash the expression (in (A1) above) upon these integrals!

    Attention At this point, many students will ‘ln ln ln’ till they’ve forgotten how to integrate the 3rd term i.e. the 1/(2+x)2.

    Remember this?

    int{}{}{(ax+b)^n} dx = 1/a {(ax+b)^{n+1}}/{n+1} + c

    {} = 4 delim{[}{{1/2}ln(1+2x)}{]}^2_1 - 2delim{[}{ln(2+x)}{]}^2_1 - delim{[}{1/{2+x}}{]}^2_1
    {} = 2 delim{[}{ln 5 - ln 3}{]} - 2 delim{[}{ln 4 - ln 3}{]} - delim{[}{1/4 - 1/3}{]}
    {} = 2 ln (5/4) + 1/12

  2. Now can you try evaluating int{2}{3}{{9-4x}/{(2x+3)(x-1)^2}} dx? 😉

B. Integrating Trigonometric Functions Using FURTHER TRIGONOMETRIC IDENTITIES

Basic Rules of Integrating Trigonometric Functions

Firstly, you should all know these by now:

  1. int{}{}{cos (ax + b)} dx = 1/a sin (ax + b) + c
  2. int{}{}{sin (ax + b)} dx = -1/a cos (ax + b) + c – note the minus sign!
  3. int{}{}{sec^2 (ax + b)} dx = 1/a tan (ax + b) + c
  4. int{}{}{tan^2 (ax + b)} dx
    {}= int{}{}{delim{[}{sec^2 (ax + b) - 1}{]}} dx (∵ sec2x = 1 + tan2x)
    {} = 1/a tan (ax + b) - x + c

Starting this year, however, strange forms of trigonometric functions may be pleading for your integrating wizardry in the O-Levels e.g. int{}{}{sin^2 x} dx, int{}{}{sin x cos x} dx, int{}{}{sin 2x cos 3x} dx etc.

These are brought to you courtesy of the new Trigonometric Identities of the new syllabus, namely the Double Angle Formulae, Addition Formulae and the Factor Formulae.

While these formulae will be provided in your exam, you’re likely to find yourself frequently using the following derived Double Angle expressions to solve these new wave integration problems:

  1. sin x cos x = {1/2} sin 2x —– (B1)
  2. cos^2 x =  {1 + cos 2x}/2 —– (B2)
  3. sin^2 x =  {1 - cos 2x}/2 —–(B3)

Armed with these, we can now join the Gen-Y Integration Movement!

SAMPLE QUESTIONS

  1. int{}{}{sin^2 x} dx = int{}{}{{1 - cos 2x}/2} dx – Using (B3) above
    {} = {1/2} int{}{}{delim{[}{1 - cos 2x}{]}} dx
    {} = {1/2} (x - {1/2} sin 2x) + c

  2. int{}{}{sin x cos x} dx =  int{}{}{{1/2} sin 2x} dx – Using (B1) above
    {} = {1/2}int{}{}{sin 2x} dx
    {} = {1/2}({-1/2}cos 2x) +c = {-1/4} cos 2x +c

  3. int{}{}{sin 2x cos 3x} dx
    {} = {1/2}int{}{}{2 cos 3x sin 2x} dx
    {} = {1/2}int{}{}{delim{[}{sin 5x - sin x}{]}} dx
    (Using Factor Formula sin A sin B = 2 cos ½(A+B) sin ½(AB) )
    {} = {1/2}({-1/5}cos 5x + cos x) + c = {cos x}/2 - {cos 5x}/10 + c

Now can you evaluate these? 😉

  1. int{pi/4}{pi/2}{delim{[}{sin^2x - 4 sin x cos x}{]}} dx
  2. int{}{}{(sin x - cos x)^2} dx

P.S. Check out this online integrator for quick answers to your integration problems.

Attention*LAST BUT NOT LEAST Miss Loi is too gentle and demure to even harm an ant, much less hurl big Shing Lee textbooks at daydreaming students.

頑張って!!!

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

  1. Leave A Comment!
Comments & Reactions

5 Comments

  1. Mgccl's Avatar
    Mgccl commented in tuition class


    2008
    Oct
    19
    Sun
    10:52am
     
    1

    Integration... me don't like...
    It seems you are doing well...
    I'm working hard to prepare for math contests...
    Cya next year 🙂

  2. H2SO4's Avatar
    H2SO4 commented in tuition class


    2008
    Oct
    20
    Mon
    12:06am
     
    2

    intergration. oh man.
    eeeeeeeeeeeeeeee. lols.

  3. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    2008
    Oct
    20
    Mon
    9:30am
     
    3

    We now have evidence that the disdain for Integration is a global phenomenon.

    Mgccl: Next year?! Now's only October! What super duper math contests are those? You need to solve for the Meaning of Life?

  4. sarah's Avatar
    sarah commented in tuition class


    2009
    Jun
    2
    Tue
    3:11pm
     
    4

    i never understood integration well till i read your notes :p

  5. TJK's Avatar
    TJK commented in tuition class


    2010
    May
    7
    Fri
    8:36pm
     
    5

    implicit differentiation and subsequently integration ???

    Anyway, GC now can help solve very hard differentiation and integration which i struggle in the past

    GC roxs =)

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