Posted by gerrard on Saturday, May 21, 2011 at 1:43pm.
An interesting problem! In fact, it would make a good exam question, because you need to know four rules to calculate the integral, or equivalent to four short questions.
(cos(4x)+2x^2)(sin(4x)-x)
=cos(4x)sin(4x)-xcos(4x)+2x²sin(4x)-2x³
1. cos(4x)sin(4x)
use substitution
u=sin(4x)
du=4cos(4x)dx
so
∫cos(4x)sin(4x)dx
=(1/4)∫udu (use the power rule)
2. -xcos(4x)
Use integration by parts:
∫-xcos(4x)dx
=-(1/4)xsin(4x)+(1/4)∫sin(4x)
3. 2x²sin(4x)
Use integration by parts, similar to (2) above.
4. -2x³
use the power rule.
Post your answer for a check if you wish. Do not forget the constant of integration (C).
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