Lecture3

# Lecture3 - Lecture 3 and 4 MATH 138-W12-003 January 9 I...

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Lecture 3 and 4: MATH 138-W12-003 January 9 and 11, 2012 I) Trigonometric integrals, Section 7.2 Example: How would we evaluate R sin 3 x dx ? Based on our previous example we could try and separate it as u = sin x and use the identity sin 2 x = 1 - cos 2 x . This yields Z sin 3 x dx = Z (1 - cos 2 x ) sin x dx, = Z (1 - cos 2 x ) sin x dx, = Z sin x dx - Z cos 2 x sin x dx. The ﬁrst is easily evaluated directly and the second can be evaluated using the substitution u = cos x . In general if we have something of the form R sin n +1 x cos m dx where n and m are even we can play the same game, by rewriting sin n using our identity sin 2 x = 1 - cos 2 x and then substituting u = cos x . This works even if m = 0 . What if we want to integrate R sin m x dx with m even? Example Evaluate R sin 2 x dx . We have to recall another identity that states sin 2 x = 1 2 (1 - cos 2 x ) Z sin 2 x dx = 1 2 Z (1 - cos(2 x )) dx The ﬁrst integral is obvious the second requires a substitution u = 2 x . Example

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## This note was uploaded on 02/14/2012 for the course MATH 138 taught by Professor Anoymous during the Fall '07 term at Waterloo.

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Lecture3 - Lecture 3 and 4 MATH 138-W12-003 January 9 I...

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