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Calculus II                                                        Quiz #6                                   April 6, 2004

Name____________________                  R.  Hammack                              Score ______
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(1)   ∫tan^3(x) sec^3(x) dx =  ∫tan^2(x) sec^2(x)   sec(x) tan(x) dx =

∫ (sec^2(x) - 1) sec^2(x)   sec(x) tan(x) dx = ∫ (u^2 - 1) u^2du = ∫u^4 - u^2du = u^5/5 - u^3/3 + C

= sec^5(x)/5 - sec^3(x)/3 + C


(2)      ∫cos^3(x) sin^3(x) dx =  ∫cos^2(x) sin^3(x) cos(x) dx =

∫ (1 - sin^2(x)) sin^3(x) cos(x) dx = ∫ (1 - u^2) u^3du = ∫ (u^3 - u^5) du = u^4/4 - u^6/6 + C

= sin^4(x)/4 - sin^6(x)/6 + C



(3)   ∫cos^5(x) dx =  ∫cos^4(x) cos(x) dx = ∫ (cos^2(x))^2cos(x) dx =   

∫ (1 - sin^2(x))^2cos(x) dx = ∫ (1 - u^2)^2du =   ∫ (1 - 2u^2 + u^4) du = u - (2u^3)/3 + u^5/5 + C

= sin(x) - (2sin^3(x))/3 + sin^5(x)/5 + C