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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 26 18.01 Fall 2006 Lecture 26: Trigonometric Integrals and Substitution Trigonometric Integrals How do you integrate an expression like sin n x cos m xdx ? ( n = 0 , 1 , 2 ... and m = 0 , 1 , 2 ,... ) We already know that: sin xdx = cos x + c and cos xdx = sin x + c Method A Suppose either n or m is odd. Example 1. sin 3 x cos 2 xdx . Our strategy is to use sin 2 x + cos 2 x = 1 to rewrite our integral in the form: sin 3 x cos 2 xdx = f (cos x ) sin xdx Indeed, sin 3 x cos 2 xdx = sin 2 x cos 2 x sin xdx = (1 cos 2 x ) cos 2 x sin xdx Next, use the substitution u = cos x and du = sin xdx Then, (1 cos 2 x ) cos 2 x sin xdx = (1 u 2 ) u 2 ( du ) 1 1 1 1 = ( u 2 + u 4 ) du = 3 u 3 + 5 u 5 + c = 3 cos 3 u + 5 cos 5 x + c Example 2. cos 3 xdx = f (sin x ) cos xdx = (1 sin 2 x ) cos xdx Again, use a substitution, namely u = sin x and du = cos xdx u 3 sin 3 x cos 3 xdx = (1 u 2 ) du = u + c = sin x + c 3 3 1...
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lec26 - MIT OpenCourseWare http://ocw.mit.edu 18.01 Single...

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