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Unformatted text preview: Z π 1 2 (cos x +  cos x  ) dx. 1 Answer: For x between 0 and π/ 2, cos x is nonnegative, so  cos x  = cos x . On the other hand, cos x ≤ 0 for x between π/ 2 and π , so  cos x  =cos x on [ π/ 2 ,π ]. Therefore, using property 5 of the deﬁnite integral, Z π 1 2 (cos x +  cos x  ) dx = Z π/ 2 1 2 (cos x +  cos x  ) dx + Z π π/ 2 1 2 (cos x +  cos x  ) dx = Z π/ 2 1 2 (cos x + cos x ) dx + Z π π/ 2 1 2 (cos xcos x ) dx = Z π/ 2 cos xdx + Z π π/ 2 dx. The second term is zero, so we just need to compute R π/ 2 cos xdx . Using the Fundamental Theorem of Calculus, since sin x is an antiderivative of cos x , we know that Z π/ 2 cos xdx = [sin x ] π/ 2 = sin( π/ 2)sin(0) = 10 = 1 . Putting this all together, then, we see that R π 1 2 (cos x +  cos x  ) dx = 1. 2...
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This note was uploaded on 01/18/2012 for the course MATH 2250 taught by Professor Chestkofsky during the Fall '08 term at University of Georgia Athens.
 Fall '08
 CHESTKOFSKY
 Calculus, Geometry, Fundamental Theorem Of Calculus

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