7.3 Some Trigonometric Integrals.pdf
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1、 Some Trigonometric Integrals How to compute these integrations? sin n xdx cos n xdx sincosmxnxdx coscosmxnxdx sinsinmxnxdx sincos mn xxdx Basic Identities Pythagorean Identities 22 sincos1xx 22 1tansecxx 22 1cotcscxx Half-Angle Identities 2 1cos2 sin 2 x x 2 1cos2 cos 2 x x Basic Identities Product
2、 Identities 1 (1)sincossin()sin() 2 mxnxmn xmn x 1 (2)sinsincos()cos() 2 mxnxmn xmn x 1 (3)coscoscos()cos() 2 mxnxmn xmn x 5 sin ( odd)xdx n 4 sinsinxxdx Find Example 1 35 21 coscoscos 35 xxxC 24 (12coscos) cosxx dx Polynomial of 22 (1cos)cosx dx 4 sincosxdx cosx 1 cos2 2 x dx Find Example 2 2 sin (
3、 even)xdx n 11 sin2 24 xxC Half-Angle Identity 1 (1 cos2 ) 2 x dx Find Example 3 24 sincos (Both and even)xxdxmn 2 1cos21cos2 22 xx dx 2 111 cos4sin 2 cos2 822 xxx dx 3 1 111 sin4sin 2 8 286 xxxC Half-Angle Identities Find Example 4 sin2 cos3xxdx 1 sin(5 )sin() 2 xx dx Product Identity 11 sin55sin 1
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