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# Homework 1 - 2 tan x t will convert any rational function...

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HOMEWORK 1 (Math 200 A, B) 1. Establish the following reduction formulas: (20 pts) (a) dx x n n n x x dx x n n n 2 2 sec 1 2 1 tan sec sec (b) dx x n x dx x n n n 2 1 tan 1 tan tan ( Hint : Write x n sec as x x n 2 2 sec sec and make use of the identity 1 cos sin 2 2 x x . Same for x n tan ). 2. (a) Given that n and m are positive integers, show the following: (20 pts) (i) 0 cos sin dx mx nx (ii) m n if m n if dx mx nx , , 0 sin sin (iii) m n if m n if dx mx nx , , 0 cos cos (b) The ( finite ) Fourier Series of ) ( x f is given by the sum: Nx a x a x a kx a x f N N k k sin ... 2 sin sin sin ) ( 2 1 1 Show that the n th coefficient n a is given by the formula: nxdx x f a n sin ) ( 1 ( Hint : Make use of the product identities for part (a)).

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3. Evaluate the integral: 2 / 3 2 2 ] ) [( b ax dx (10 pts) 4. The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution
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Unformatted text preview: ) 2 / tan( x t will convert any rational function of x sin and x cos into an ordinary rational function of t . (40 pts) (a) If x x t , ) 2 / tan( , show that : 2 1 ) 2 / sin( t t x and 2 1 1 ) 2 / cos( t x (b) Show that: 2 1 2 sin t t x and 2 2 1 1 cos t t x (c) Show that: dt t dx 2 1 2 (d) Use the above substitution to evaluate the integral: x dx sin 5 3 5. Evaluate the improper integral: dx x x ) 1 ( 1 (10 pts) ( Note : The above integral is of both types, type 1 and type 2)....
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Homework 1 - 2 tan x t will convert any rational function...

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