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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) ( * ) π 2 0 e 3 x cos( x ) dx (b) ( * ) t 3 e - t 2 dt (c) ( * ) e 1 In 3 x dx (d) ( * ) 1 0 x 5 e x 2 dx (e) π 0 e cos t sin(2 t ) dt (f) π 2 0 x sin( x ) dx (g) x In x dx . (h) sin 2 ( θ ) cos 3 ( θ ) . 2. (a) Prove that, if n 2 is an integer, then: sin n x dx = - 1 n cos x sin n - 1 x + n - 1 n sin n - 2 x dx. (b) Prove that, if n 2 is an integer, then: π/ 2 0 sin n x dx = n - 1 n π/ 2 0 sin n - 2 x dx. (c) Evaluate π/ 2 0 sin 3 x dx and π/ 2 0 sin 5 x dx (d) Show that 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 tan 3 x dx . (b) ( * ) Evaluate tan 4 ( x ) sec 4 ( x ) dx . (c) Evaluate sin(2 x ) cos( x ) dx , sin(2 x ) sin(3 x ) dx , cos(2 x ) cos(4 x ) dx .
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