[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