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integral operation

# integral operation - x sin(2ax 2 4a 3cos(ax cos(3ax 3...

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2 3 2 3 2 2 2 2 sin(2 ) sin ( ) 2 4 3cos( ) cos(3 ) sin ( ) 4 12 sin(2 ) cos ( ) 2 4 3sin( ) sin(3 ) cos ( ) 4 12 [cos( )] sin( )cos( ) 2 1 sin ( )cos ( ) sin(4 ) 32 8 sin ( )cos( ) x ax ax dx a ax ax ax a a x ax ax dx a ax ax ax dx a a ax ax ax dx a x ax ax dx ax a ax ax dx = = + = + = + = = + 3 3 2 1 1 [sin( )] 3 [cos( )] sin( )cos ( ) 3 [sin( )] sin ( )cos( ) where n is a positive integer ( 1) [cos( )] sin( )cos ( ) where n is a positive integer ( 1) sin( ) n n n n ax a ax ax ax dx a ax ax ax dx n a ax ax ax dx n a kx e x + + = = = + = + 0 2 2 0 2 2 2 0 2 3 2 2 2 3 0 2 2 2 2 3 3 3 2 2 3 4 4 4 arctan sin( ) 2 sin( ) ( ) 6 2 sin( ) ( ) ( 1) ( 2 2) ( 3 6 6) ( ax ax ax ax ax ax ax ax ax ax ax ax k dx a k kx e dx a k ak x kx e dx a k a k k x kx e dx a k e ax xe dx a e a x ax x e dx a e a x a x ax x e dx a e a x e dx = = + = + = + = + = + = = 4 3 3 2 2 5 4 12 24 24) sin( ) sin cos cos sin cos( ) cos cos sin sin x a x a x ax a x y x y x y x y x y x y + + ± = ± ± = m

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