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M138Assign1W06wMaple

# M138Assign1W06wMaple - MATH 138 Assignment 1 Winter 2006...

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MATH 138 Assignment 1 Winter 2006 Submit all problems marked * on Friday, January 13, plus the questions from the Maple Addendum. 1. Evaluate the following antiderivatives and definite integrals, using a suitable change of vari- able, indentity, 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 *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 2. Find the partial fractions decomposition of each function, and hence find f ( x ) dx . a) f ( x ) = 10 x + 2 x 3 - 5 x 2 + x - 5 *b) f ( x ) = x + 1 x 3 + x c) f ( x ) = x x 2 - 3 x + 2 d) f ( x ) = x 2 x 2 + 25 *e) f ( x ) = x - 2 x 2 - x 4 *f) f ( x ) = x 4 + 3 x 3 + 2 x 2 + 1 x 2 + 3 x + 2 3. *a) Find constants A and B such that d dx ( e ax ( A sin bx + B cos bx )) = e ax sin bx for all x, where a and b are given constants. *b) Hence find the general antiderivative e ax sin bx dx . *c) Evaluate π 0 e - x sin x dx . d) Evaluate t 0 e s sin( t - s ) ds e) Verify your result in b) by doing two integrations by parts. 4. Use integration by parts to prove each reduction formula. a) I n = cos n x dx I n = 1 n cos n - 1 x sin x + n - 1 n I n - 2 b) I n = (ln x ) n dx I n = x (ln x ) n - nI n - 1

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5. a) Find the area of the region bounded by y = 0, and y = xe - x , for 0 x b . What happens to this area as b + ? *b) Find the area enclosed by y = ln x, y = 1 x - 1, and x = e . *6. A rocket lifts off at t = 0 and has velocity v ( t ) = - 9 . 8 t - 3000 ln ( 1 - 2 375 t ) metres per second. Find the height of the rocket 1 minute after lift off. 7. Sketch a graph of each curve, develop a definite integral representing its length (as in Example 1 on page 17 of your Course Notes), and then evaluate the integral to find the length. a) the curve y = 1 6 x 3 + 1 2 x for 1 x 3 ; *b) the closed curve x 2 / 3 + y 2 / 3 = 1 ; c) the catenary y = 1 2 ( e x + e - x ) for - 1 x 1 . d) the curve y = ln(cos x ) for 0 x π/ 4.
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M138Assign1W06wMaple - MATH 138 Assignment 1 Winter 2006...

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