7.3 Some Trigonometric Integrals.pdf

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 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 ( 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 102 xd xxdx 11 cos5cos 102 xxC Summary Pythagorean Identities 22 sincos1xx 22 1tansecxx 22 1cotcscxx Half-Angle Identities 2 1cos2 sin 2 x x 2 1cos2 cos 2 x x Summary Product 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 Summary sincos mn xxdx When m, n has an odd number, look for a substitution; when m, n are both even number, use half-angle identities. cos ( sin )xx dx Questions and Answers sin cosxxdx 2 1 cos 2 xC substitution Q1:Method1 coscosxdx sin (cos )xx dx Questions and Answers sin cosxxdx 2 1 sin 2 xC Q1: Method2 sinsinxdx substitution Questions and Answers sinsin( and are positive integers)mxnxdx mn 1 cos()cos() 2 mn xmn x dx 0 Case1:mn 111 sin()sin() 2 mn xmn x mnmn Q2: Questions and Answers 1 cos21 2 mxdx Case2:mn 11 sin2 2 2 mxx m sinsin( and are positive integers)mxnxdx mn Q2: Some Trigonometric Integrals