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Unformatted text preview: MATH 118, LECTURE 3: Trigonometric Integrals 1 Trigonometric Integrals Many trigonometric integrals can be solved with simple manipulation. We will break the problems for this lecture into three sections: • integrals involving sin( x ) and cos( x ), • integrals involving tan( x ) and sec( x ), and • integrals involving cot( x ) and csc( x ). The reason for this grouping is that differentiating the functions in these three classes still gives functions within the same grouping (e.g. differenti- ating sin( x ) or cos( x ) gives functions consisting of sin( x ) and cos( x ); differ- entiating tan( x ) and sec( x ) gives functions consisting of tan( x ) and sec( x ); etc.). In this lecture, we will make use of this fact to help us integrate functions containing elements within these groupings. 1.1 Integrals with sin( x ) and cos( x ) Consider the derivative of powers of sin( x ) and cos( x ), e.g. d dx sin n +1 ( x ) = ( n + 1) sin n ( x ) cos( x ) . We can rearrange the constants and integrate to get integraldisplay sin n ( x ) cos( x ) dx = 1 n + 1 sin n +1 ( x ) + C,n negationslash =- 1 . (1) Similarly, for powers of cos( x ) we have integraldisplay cos n ( x ) sin( x ) dx =- 1 n + 1 cos n +1 ( x ) + C,n negationslash =- 1 . (2) These integrals could also be derived by integration by substitution, choosing u = sin( x ) and u = cos( x ) respectively (try it!). 1 In other words, if we can manipulate integrals containing sin( x ) and cos( x ) so that there is a product involving a single sin( x ) or cos( x ) term, we can integrate the expression. We can handle the case when n =- 1 by substitution to complete (1) and (2): integraldisplay sin- 1 ( x ) cos( x ) dx = integraldisplay cot( x ) dx = ln | sin( x ) | + C, integraldisplay cos- 1 ( x ) sin( x ) dx = integraldisplay tan( x ) dx = ln | sec( x ) | + C....
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- Spring '09
- Cos, dx