138ass1 - MATH 138 Assignment 1 Spring 2009 Methods of...

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MATH 138 Assignment 1 Spring 2009 Methods of integration and applications: Integration by parts, substitution, trigonometric substitution, area between curves, average value of a function. Submit all problems marked (*) by 12:30 p.m. on Friday, May 15. 1. Evaluate the following antiderivatives and definite integrals. (a) ( * ) R π 2 0 e 3 x cos( x ) dx (b) ( * ) R t 3 e - t 2 dt (c) ( * ) R e 1 In 3 x dx (d) ( * ) R 1 0 x 5 e x 2 dx (e) R π 0 e cos t sin(2 t ) dt (f) R π 2 0 x sin( x ) dx (g) R x In x dx . (h) R sin 2 ( θ ) cos 3 ( θ ) . 2. (a) Prove that, if n 2 is an integer, then: Z sin n x dx = - 1 n cos x sin n - 1 x + n - 1 n Z sin n - 2 x dx. (b) Prove that, if n 2 is an integer, then: Z π/ 2 0 sin n x dx = n - 1 n Z π/ 2 0 sin n - 2 x dx. (c) Evaluate R π/ 2 0 sin 3 x dx and R π/ 2 0 sin 5 x dx (d) Show that Z π/ 2 0 sin 2 n +1 x dx = 2 · 4 · 6 . . . 2 n 3 · 5 · 7 . . . (2 n + 1) π 2 3. Take a look at Section 7.2 of Stewart if you want some help for this problem. (a) Find
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This note was uploaded on 01/19/2010 for the course MATH 138 taught by Professor Anoymous during the Spring '07 term at Waterloo.

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138ass1 - MATH 138 Assignment 1 Spring 2009 Methods of...

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