quiz2a - Solutions to quiz 2A 1 Evaluate the integral(a...

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Solutions to quiz 2A www.math.ufl.edu/˜mathguy 1. Evaluate the integral. (a) R cos 4 ( θ ) (Solution) We will use the fact that cos 2 ( θ ) = 1 2 (1 + cos(2 θ )). Thus, Z cos 4 ( θ ) = Z (cos 2 ( θ )) 2 = Z 1 2 (1 + cos(2 θ )) 2 = 1 4 Z 1 + 2 cos(2 θ ) + cos 2 (2 θ ) = 1 4 Z 1 + 2 Z cos (2 θ ) + Z cos 2 (2 θ ) ! = 1 4 θ + sin(2 θ ) + 1 2 Z (1 + cos(4 θ )) ! = 1 4 θ + sin(2 θ ) + 1 2 θ + 1 8 sin(4 θ ) + C ! = 3 8 θ + 1 4 sin(2 θ ) + 1 32 sin(4 θ ) + C 1 (b) R e x sin( x ) dx (Solution) Here we will use integration by parts. u = sin( x ) dv = e x dx du = cos( x ) dx v = e x Z e x sin( x ) dx = Z udv = u · v - Z vdu = sin( x ) e x - Z e x cos( x ) dx Now, we will use integration by parts on R e x cos( x ) dx . u 1 = cos( x ) dv 1 = e x dx du 1 = - sin( x ) dx v 1 = e x Z
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