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secx - MATH 112-SPRING 2008 Section 5 Achilleas...

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MATH 112-SPRING 2008 Section 5 Achilleas Sinefakopoulos [email protected] FOUR WAYS TO FIND Z sec x dx Below we outline four ways to compute Z sec x dx . As an exercise, it is recommended to complete the details left out by yourselves. 1. 1st way : Write sec x = sec x · (sec x + tan x ) sec x + tan x = sec 2 x + sec x tan x sec x + tan x . Now apply the substitution u = sec x + tan x to get Z sec x dx = ln | sec x + tan x | + C. 2. 2nd way : First write sec x = 1 cos x = cos x cos 2 x = cos x 1 - sin 2 x = 1 2 cos x 1 + sin x + cos x 1 - sin x an expression obviously inspired by partial fraction decomposition. Be sure to note that Z cos x 1 - sin x dx = - ln | 1 - sin x | ; the minus sign is very important. And remember that 1 2 ln a = ln a . From there on, keep doing algebra, and trust to luck. 3. 3rd way :Apply the world’s sneakiest substitution t = tan x 2 , x = 2 tan - 1 ( t ) , dx = 2 1 + t 2 dt. Check that sin x = 2 t 1 + t 2 , cos x = 1 - t 2 1 + t 2 . Once again, quite a bit of manipulation is required to put the answer in the desired form; the expression tan x 2
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