trigin_hnd - Trigonometric Integrals Guidelines for...

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Trigonometric Integrals Guidelines for Evaluating A. ( 29 ( 29 dx x x n m cos sin m n 0 0 and are integers 1. m is an odd integer ( 29 ( 29 ( 29 ( 29 ( 29 - = dx x x x dx x x n m n m sin cos sin cos sin 1 express ( 29 ( 29 ( 29 x x x m 2 2 1 cos 1 sin : sin - = - substitute: ( 29 ( 29 dx x du x u sin , cos - = = 2. n is an odd integer ( 29 ( 29 ( 29 ( 29 ( 29 - = dx x x x dx x x n m n m cos cos sin cos sin 1 express ( 29 ( 29 ( 29 x x x n 2 2 1 sin 1 cos : cos - = - substitute: ( 29 ( 29 dx x du x u cos , sin = = 3. m and n are even Use half-angle formulas to reduce the exponents by one-half. ( 29 ( 29 2 2 cos 1 sin 2 x x - = ( 29 ( 29 2 2 cos 1 cos 2 x x + = B. ( 29 ( 29 dx x x n m sec tan m n 0 0 and are integers 1. m is an odd integer ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - = dx x x x x dx x x n m n m tan sec sec tan sec tan 1 1 express ( 29 ( 29 ( 29 1 sec tan : tan 2 2 1 - = - x x x m substitute: ( 29 ( 29 ( 29 dx x x du x u tan sec , sec = = 2. n is an even integer ( 29 ( 29 ( 29 ( 29 ( 29 - = dx x x x dx x x n m n m 2 2 sec sec tan sec tan express ( 29 ( 29 ( 29 x x x n 2 2 2 tan 1 sec : sec + = - substitute: ( 29 ( 29 dx x du x u 2 sec , tan = = 3. m is even and n is odd There is no standard method of evaluation. Possibly use integration by parts. C. ( 29 ( 29 dx x x n m csc cot m n 0 0 and are integers
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1. m is an odd integer ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - = dx x x x x dx x x n m n m csc cot csc cot
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Unformatted text preview: ( 29 ( 29 ( 29 1 csc cot : cot 2 2 1-=-x x x m substitute: ( 29 ( 29 ( 29 dx x x du x u cot csc , csc-= = 2. n is an even integer ( 29 ( 29 ( 29 ( 29 ( 29 ∫ ∫-= dx x x x dx x x n m n m 2 2 csc csc cot csc cot express ( 29 ( 29 ( 29 x x x n 2 2 2 cot 1 csc : csc + =-substitute: ( 29 ( 29 dx x du x u 2 csc , cot-= = 3. m is even and n is odd There is no standard method of evaluation. Possibly use integration by parts. D. ( 29 ( 29 ( 29 [ ] ∫ ∫-+ + = dx nx mx nx mx dx nx mx ) cos( cos 2 1 cos cos ( 29 ( 29 [ ] ∫ ∫ +--= dx nx mx nx mx dx nx mx ) cos( ) cos( 2 1 sin sin ( 29 ( 29 ( 29 [ ] ∫ ∫-+ + = dx nx mx nx mx dx nx mx ) sin( sin 2 1 cos sin...
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