� sin4 (x) cos2 (x) dx � Compute sin4 (x) cos2 (x) dx. Solution Because all of the exponents in this problem are even, our chosen solution in volves half angle formulas: 1 + cos(2θ) 2 1 − cos(2θ) 2 sin θ = . 2 cos2 θ = Because we have to do a lot of writing before we actually integrate anything, we’ll start with some “side work” to convert the integrand into something we know how to integrate. sin4 x cos2 x = = = = = (sin2 x)2 cos2 x � �2 � � 1 − cos(2x) 1 + cos(2x) 2 2 � �� 2 1 − 2 cos(2x) + cos (2x) 1 + cos(2x) 4 2 2 1 − 2 cos(2x) + cos (2x) + cos(2x) − 2 cos2 (2x) + cos3 (2x) 8 1 − cos(2x) − cos2 (2x) + cos3 (2x) 8 This is all the side work we need to do here, because we know that: � x sin(2x) cos2 (2x) = + + c1 and 2 4 � 1 1 cos3 (2x) = sin(2x) − sin3 (2x) + c2 . 2 6 We conclude that: � � 1 − cos(2x) − cos2 (2x) + cos3 (2x) 4 2 sin x cos x dx = dx 8 � � � 1 1 x sin(2x) = x − sin(2x) − + + c1 8 2 2 4 � �� 1 1 + sin(2x) − sin3 (2x) + c2 2 6 � � 1 x sin(2x) 1 3 − − sin (2x) + C = 8 2 4 6 = x sin(2x) sin3 (2x) − − +C 16 32 48 It’s diﬃcult to check that this is the correct answer. If C = 0 this is an odd function which is at least consistent with the integrand being an even function.