Monday, 21 April 2014

real analysis - Evaluate intcos2xsin4xmathrmdx





Evaluate integral cos2xsin4xdx.




Attempt. Setting tanx=t, gives:
cos2xsin4xdx=11+t2(t21+t2)2dt1+t2=t4(1+t2)4dt,


which does not seem to be elementary.




Thank in advance for the help.


Answer



Here is to integrate economically,



cos2xsin4x=18sin22x(1cos2x)=116116cos4x18sin22xcos2x



Thus,



cos2xsin4xdx=x16164sin4x148sin32x+C


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