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MATH 2202 Quiz 1 Solution: Indefinite Integrals with Substitution and Differentiation - Pr, Quizzes of Calculus

The solution to quiz 1, version 2, of math 2202, which involves evaluating an indefinite integral using substitution and checking the answer through differentiation.

Typology: Quizzes

2010/2011

Uploaded on 06/03/2011

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MATH 2202 –Quiz 1 (Version 2) Solution
January 16, 2008
NAME_______________________________________
1. Use a substitution to evaluate the inde…nite integral
Zsin (2x)dx.
2. Use di¤erentiation to check whether or not the answer you obtained
in part 1 is correct. Have you found that your answer is correct or
incorrect?
You must show your procedures in order to receive any credit. You will
not receive credit if you just write down an answer without showing how that
answer was obtained.
Solution
1. To evaluate this inde…nite integral, we use the substitution
u= 2x
du = 2 dx.
We then evaluate the integral as follows:
Zsin (2x)dx =1
2Zsin (u)du =1
2(cos (u)) + C=1
2cos (2x) + C.
2. Show the check of your answer here. Do you nd that your answer is
correct or incorrect? (Please answer this in a complete sentence.)
d
dx 1
2cos (2x)=1
2(sin (2x)) (2) = sin (2x)
shows that our answer is correct.
1

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MATH 2202 ñQuiz 1 (Version 2) Solution January 16, 2008

NAME_______________________________________

  1. Use a substitution to evaluate the indeÖnite integral Z sin (2x) dx.
  2. Use di§erentiation to check whether or not the answer you obtained in part 1 is correct. Have you found that your answer is correct or incorrect?

You must show your procedures in order to receive any credit. You will not receive credit if you just write down an answer without showing how that answer was obtained.

Solution

  1. To evaluate this indeÖnite integral, we use the substitution

u = 2x

du = 2 dx. We then evaluate the integral as follows: Z sin (2x) dx =

Z

sin (u) du =

( cos (u)) + C =

cos (2x) + C.

  1. Show the check of your answer here. Do you Önd that your answer is correct or incorrect? (Please answer this in a complete sentence.)

d dx

cos (2x)

 ( sin (2x))  (2) = sin (2x)

shows that our answer is correct.