A. evaluate ∫20 tan 2x sec^2 2x dx using the substitution u = tan 2x.
b. evaluate ∫20 tan 2x sec^2 2x dx using the substitution u = sec 2x.
c. reconcile the results in parts (a) and (b).

The integral is equal to for an arbitrary constant C.

Step-by-step explanation:

a) If then so the integral becomes . (the constant of integration is actually 5C, but this doesn't affect the result when taking derivatives, so we still denote it by C)

b) In this case hence . We rewrite the integral as .

c) We use the trigonometric identity is part b). The value of the integral is . which coincides with part a)

Note that we just replaced 5+C by C. This is because we are asked for an indefinite integral. Each value of C defines a unique antiderivative, but we are not interested in specific values of C as this integral is the family of all antiderivatives. Part a) and b) don't coincide for specific values of C (they would if we were working with a definite integral), but they do represent the same family of functions.  

answer: a

step-by-step explanation:

if xy ~ yz therefore xy would be the same length as yz