Math118_S10_W2

Math118_S10_W2 - Math118(M Kohandel 1 Outline(week 2 2.1 Trigonometric integrals Integrals involving sine and cosine(8.3 Integral involving tangent

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Unformatted text preview: Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 2) 2.1 Trigonometric integrals Integrals involving sine and cosine (8.3) Integral involving tangent and secant (8.3) 2.2 Trigonometric substitution (8.4) 2.3 Completing the square (8.5) Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 2.1 Trigonometric integrals Integrals involving sine and cosine: We already know how to evaluate integrals nxdx sin , nxdx cos , xdx x n cos sin (change of variable x u sin ) and xdx x n sin cos (change of variable x u cos ). Here, we evaluate integrals of the more general form xdx x n m cos sin . Case 1: If m or n is odd: (a) split off the factor with the odd power, (b) use the identity 1 cos sin 2 2 x x , and (c) integrate using the change of variable; see the examples below. For example if ) odd ( 1 2 k m , then du u u I x u xdx x x xdx x x xdx x I n k n k n k n k ) 1 ( cos sin cos ) cos 1 ( sin cos ) (sin cos sin 2 2 2 1 2 Example 2.1: Evaluate xdx x 2 5 cos sin C x x x C u u u du u u u du u u I xdx du x u xdx x x xdx x x xdx x I 7 cos 5 cos 2 3 cos 7 5 2 3 ) 2 ( ) 1 ( sin cos cos ) cos 1 ( sin cos sin sin cos sin 7 5 3 7 5 3 6 4 2 2 2 2 2 2 2 2 4 2 5 Example 2.2: Evaluate xdx 3 cos ) cos sin ( 3 sin sin 3 ) 1 ( ) sin 1 ( cos cos cos cos 3 3 2 2 2 3 xdx du x u C x x C u u du u dx x x xdx x xdx Case 2: If m and n are both even, we use the double angle identities ) 2 cos 1 ( 2 1 cos , ) 2 cos 1 ( 2 1 sin , 2 sin 2 1 cos sin 2 2 x x x x x x x Example 2.3: Evaluate (a) xdx 2 sin and (b)...
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This note was uploaded on 02/06/2011 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.

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Math118_S10_W2 - Math118(M Kohandel 1 Outline(week 2 2.1 Trigonometric integrals Integrals involving sine and cosine(8.3 Integral involving tangent

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