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Calculus  II                                                                Quiz #6                                              April 2, 2003

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

(a) ∫tan(x^3) x^2  dx = 1/3∫tan(x^3) 3x^2  dx = 1/3∫tan(u)   du = -1/3ln | cos(u) | +C = -1/3ln | cos(x^3) | +C

u = x^3
du = 3x^2


(b) ∫1/(4 + 3x^2)   dx = ∫1/(2^2 + (3^(1/2) x)^2)   dx = 1/3^(1/2) ∫1/(2^2 + (3^(1/2) x)^2) 3^(1/2)   dx =  1/3^(1/2) ∫1/(2^2 + (u)^2)   du = 1/3^(1/2) 1/2   tan^(-1)(u/2) = 1/(23^(1/2))    tan^(-1)((3^(1/2) x)/2)


u = 3^(1/2) x
du = 3^(1/2) dx



(c) ∫x^2ln (x)   dx =ln(x) x^3/2 - ∫x^3/3 1/xdx =ln(x) x^3/3 - ∫x^2/3 dx =ln(x) x^3/3 - x^3/9 + C

u = ln(x)             dv = x^2dx
du = 1/x dx         v = x^3/3



(d) ∫x e^(5x)   dx =xe^(5x)/5 - ∫e^(5x)/5dx= (x e^(5x))/5 - e^(5x)/25 + C

u = x               dv = e^(5x) dx
du = dx         v = e^(5x)/5