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Unformatted text preview: B U Department of Mathematics Math 101 Calculus I Spring 2005 Second Midterm Calculus archive is a property of Bo˘gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Evaluate the integrals below: (a)  Z e 4 e dx x √ ln x Solution: Let I = Z e 4 e dx x √ ln x ln x = u, dx x = du x = e ⇒ u = 1 ,x = e 4 ⇒ u = 4 ⇒ I = Z 4 1 du √ u = 2 √ u ] 4 1 = 2 . (b)  Z sin2 θ sin 2 θ + 1 dθ Solution: sin 2 θ + 1 = u ⇒ 2sin θ cos θdθ = sin2 θdθ = du ⇒ Z sin2 θ sin 2 θ + 1 dθ = Z du u = ln | u | + C = ln | sin 2 θ + 1 | + C = ln( sin 2 θ + 1) + C since sin 2 θ + 1 > 2.  A page has an area of 90cm...
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This note was uploaded on 11/16/2011 for the course MATH 102 taught by Professor Soysal during the Winter '09 term at Boğaziçi University.
- Winter '09