M138.W09.assmt1 - MATH 138 Assignment 1(2 pages Winter 2009...

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MATH 138 Assignment 1 (2 pages) Winter 2009 Submit all problems marked * by 8:20 a.m. on Friday, January 16. 1. Evaluate the following antiderivatives and definite integrals, using a suitable change of variable, identity, or integration by parts. *a) x 4 ln x dx b) 1 0 ln( x + 1) dx c) x 2 sin x dx *d) π/ 3 0 (1 - tan 2 θ ) *e) sec θ dθ f) 3 - 1 arctan x dx *g) arcsin(2 x ) dx *h) ln 3 0 x 2 e - x dx i) 9 4 1 x - 1 dx j) 1 - 1 t 2 t 2 - 2 dt *k) ln x x 2 dx l) 1 0 x arctan x dx *m) 3 1 arctan ( 1 x ) dx n) cos x dx *o) x 3 e x 2 dx *p) π/ 3 0 1 sin x - 1 dx q) x x 2 - 9 dx *r) 2 0 x 3 4 - x 2 dx s) 2 0 x 3 x 2 + 4 dx *t) x 2 ( x 2 + a 2 ) 3 / 2 dx u) x ln x dx v) sin ln x dx *w) 1 0 x 3 4 + x 2 dx *x) (ln x ) 2 dx hints : For part e), multiply sec θ by sec θ + tan θ tan θ + sec θ ; for part o), w), think x 3 = x 2 · x. *2. a) Let I = e ax sin bx dx . Use two successive integrations by parts to show that I = e ax ( a sin bx - b cos bx ) a 2 + b 2 + C. b) Evaluate t 0 e s sin( t - s ) ds . 3. Use integration by parts to prove each reduction formula, given that n is a positive integer.
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