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Calculus  II                                                      Quiz #7                                                      April 9, 2003

Name____________________                 R.  Hammack                                                 Score ______
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(1) Evaluate the following integrals.

(a) ∫tan^5(x) sec^4(x) dx =   ∫tan^5(x) sec^2(x) sec^2(x) dx = ∫tan^5(x) (1 + tan^2(x)) sec^2(x) dx = ∫u^5(1 + u^2) du = ∫u^5 + u^7du = u^6/6 + u^8/8 + C = tan^6(x)/6 + tan^8(x)/8 + C

u = tan x
du = sec^2(x) dx


(b) ∫1/(x^2 (4 - x^2)^(1/2)) dx =∫ (2cos θ)/((2sin θ)^2 (4 - (2sin θ)^2)^(1/2)) dθ =∫ (2cos θ)/((2sin θ)^22cos θ) dθ =1/4∫1/(sin θ)^2dθ =

1/4∫csc^2 θ dθ= -1/4 cot θ  + C = -(4 - x^2)^(1/2)/(4x) + C


x = 2sin θ
dx = 2cos θ dθ




(2) Find the area under the graph of y = tan^2(x) between x = 0 and x = π/4.

∫_0^(π/4) tan^2(x) dx = ∫_0^(π/4) (sec^2(x) - 1) dx =[tan(x) - x] _0^(π/4)
= (tan(π/4) - π/4) - (tan(0) - 0) = 1 - π/4 Square units.