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trig_integrals - MATH 112-SPRING 2008 INTEGRATION OF...

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MATH 112-SPRING 2008 INTEGRATION OF TRIGONOMETRIC FUNCTIONS Achilleas Sinefakopoulos 1. Evaluation of Z sin m x dx . (a) If m = 2 k + 1 is positive and odd, then we use the identity sin 2 x + cos 2 x = 1 and the substitution u = cos x to get Z sin 2 k +1 x dx = Z (sin 2 x ) k · sin x dx = Z (1 - cos 2 x ) k · sin x dx = - Z (1 - u 2 ) k du (b) If m = 2 k is positive and even, then we use the identity sin 2 x = 1 - cos 2 x 2 to get Z sin 2 k x dx = Z (sin 2 x ) k dx = Z 1 - cos 2 x 2 k dx Remark : Another way is to use a reduction formula. 2. Evaluation of Z cos m x dx . (a) If m = 2 k + 1 is positive and odd, then we use the identity sin 2 x + cos 2 x = 1 and the substitution u = sin x to get Z cos 2 k +1 x dx = Z (cos 2 x ) k · cos x dx = Z (1 - sin 2 x ) k · cos x dx = Z (1 - u 2 ) k du (b) If m = 2 k is positive and even, then we use the identity cos 2 x = 1+cos 2 x 2 to get Z cos 2 k x dx = Z (cos 2 x ) k dx = Z 1 + cos 2 x 2 k dx Remark : Another way is to use a reduction formula as we did in class.
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