Antiderivative of tan x sec2 x | Answer Key - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2019
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145

Pages:

2
Author:

carasapokt1

Antiderivative of tan x sec2 x � Compute tan x sec2 x dx in two diﬀerent ways: a) By substituting u = tan x. b) By substituting v = sec x. c) Compare the two results. Solution � a) Compute tan x sec2 x dx by substituting u = tan x. If u = tan x then du = sec2 x dx and: � tan x sec2 x dx = = = � b) Compute � u du 1 2 u +c 2 1 tan2 x + c. 2 tan x sec2 x dx by substituting v = sec x. If v = sec x then dv = sec x tan x dx and: � � tan x sec2 x dx = sec x(tan x sec x dx) � = v dv = = 1 2 v +C 2 1 sec2 x + C. 2 c) Compare the two results. At ﬁrst glance you may think you made a mistake; it is not true that tan2 x = sec2 x. However, you can see from the graph in Figure 1 that your two answers may only diﬀer by a constant. 1 Figure 1: Graphs of tan2 x (blue) and sec2 x (red). In fact, that is the case: tan2 x = = = sin2 x cos2 x 1 − cos2 x cos2 x sec2 x − 1 1 1 1 We conclude that tan2 x = sec2 x − and so the two results are equiva 2 2 2 lent up to an added constant. Both answers are correct. 2

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