Wednesday, December 2, 2015

[AP_Calculus20151201] Integrate something with arcsecant by parts

     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.

     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.

     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.

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

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