[AP_Calculus20151201] Integrate something with arcsecant by parts

Question

Introduction

     This question is taken from the Techniques of Integration chapter of Thomas’ Calculus, 12th edition.  It looks pretty nasty in that the arcsecant is just one of those things on the fringes of teachers’ and students’ minds.

Strategy

     We apply the “Integration by Parts” Formula                                                  

with the “d(etail)” heuristic.  The expression  t  is  algebraic whereas the arcsecant expression is of the “inverse” type.  Since “a” comes before  “i”  in “d(etail)”, we choose the algebraic   expression  t  to serve as our  ^dv/_dx.  We realise that we will need the derivative of the arcsecant                                                 

for  sec^-1 x  being a cute angle ... I mean, an acute angle.

Solution

Remarks

     A slight modification of this approach is to first re-express the arcsecant as  arcsec t = arccos(¹/_t).  One needs to work out the derivative of this arccosine expression when doing the integral.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics (challenge)

* IB Mathematics HL (challenge)

* Advanced Placement (AP) Calculus BC (challenge)

* University / College calculus

* other syllabuses that involve integration and inverse trigonometric functions

* any precocious learner who loves a challenge

[H2_Expository] Integration by Parts and the “d(etail)” Heuristic

Product Rule for Integration?

     How do we integrate a product of two functions e.g. find   ò x sin x dx ?   Unlike differentiation, there are not that many general rules (e.g. the Chain Rule, the Product Rule and the Quotient Rule) that we can use for integration.  However “Integration by parts” is similar to and can be obtained from the differentiation Product Rule

Applying Integration by Parts

     But how do we use the formula?  Before you do anything, analyse and classify the functions first.  You need to choose something for the  “u”  and  something for the “dv”  or  “^dv/_dx”.  For your choice of the “^dv/_dx” part, you can use the “d(etail)” heuristic as a guide.

     “e” is for exponential functions:      e.g. e^2x, 3^x.

     “t” is for trigonometric functions:   e.g. tan x, sin 3x, cos 2x.

     “a” is for algebraic functions:         e.g. 3x³, constants, 4x – x².

     “i” is for inverse functions:             e.g. sin^-1 x, tann^-1 5x.

     “l” is for logarithmic functions:      e.g. ln x, lg x, log₂ x.

  

Here are some examples of choices for  “^dv/_dx”  and  “u” using the “d(etail)” heuristic.

Integral	Analysis	dv/dx	u
ò x sin x dx	x  is algebraic, sin x  is trigonometric, t comes before a	sin x	x
ò e-x cos x dx	e-x  is exponential, cos x  is trigonometric, e comes before t	e-x	cos x
ò x tan-1 x dx	x  is algebraic, tan-1 x  is inverse, a comes before i	x	tan-1 x
ò ln x dx = ò (ln x)(1) dx	1  is algebraic, ln x  is logarithmic, a comes before l	1	ln x

The above is actually equivalent to “liate” for the choice of  u, which is taught by many lecturers trained in American universities.  Once you have chosen  v,  the other part u  is automatically chosen,  and vice versa.  But personally, I think “d(etail)” is easier to remember: 

If you forget the details, just remember “d(etail)”!

I shall now illustrate the working of the first example with different styles of presentation.

Presentation 1 (for beginners – using “u” and “v” explicitly)

Presentation 2 (intermediate – using “pre-integration”)

With sufficient practice, the integration can be written down quickly as follows:-

Presentation 3 (advanced – for speed)

 

Remarks

     The “d(etail)” heuristic is a special one that is invented for Integration by Parts.  Like all heuristics, it is just a guideline or rule-of-thumb.  It works most of the time, but not all the time.  If you find that this does not work, you need to try different combinations of  u  and  dv.  The part chosen for “^dv/_dx” should be more easily integrable, or at least, you already know its integral.  After doing the by parts procedure, you should end up with an integral not more complicated than the original one.

Definite integrals

 

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics

* revision for IB Mathematics HL

* Advanced Placement (AP) Calculus BC

* other syllabuses that involve integration by parts

* any precocious or independent learner who loves calculus