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Calculus II                                                        Quiz #7                                   April 13, 2004

Name____________________                  R.  Hammack                              Score ______
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(1)   ∫1/(x^2 - 1)^(1/2) dx =  ∫1/(sec^2(θ) - 1)^(1/2) sec(θ) tan(θ) dθ = ∫ (sec(θ) tan(θ))/tan(θ) dθ = ∫sec(θ) dθ =

ln | sec(θ) + tan(θ) | +C = ln | x + (x^2 - 1)^(1/2) | +C

x = sec(θ)  dx = sec(θ) tan(θ) dθ     [Graphics:HTMLFiles/quiz7_5.gif]              






(2)      ∫ (4x^2 + 2x + 1)/(x^3 + x) dx =   ∫ (4x^2 + 2x + 1)/x(x^2 + 1) dx = ∫ (1/x + (3 x + 2)/(x^2 + 1)) dx

∫1/xdx + ∫ (3 x )/(x^2 + 1) dx + ∫ ( 2)/(x^2 + 1) dx = ln | x | +2/3ln | x^2 + 1 | +2tan^(-1)(x) + C



(4x^2 + 2x + 1)/x(x^2 + 1) = A/x + (B x + C)/(x^2 + 1)

4x^2 + 2x + 1 = A(x^2 + 1) + (B x + C) x

4x^2 + 2x + 1 = (A + B) x^2 + C x + A

Thus A = 1, C = 2, and B = 3,  so

(4x^2 + 2x + 1)/x(x^2 + 1) = 1/x + (3 x + 2)/(x^2 + 1)