Question #7a30e

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Answer 1 · 2017-12-28T14:18:49

#int cos^2(2x)sinx = - (12cos^5x-20cos^3x+15cosx)/15+C#

Note that:

#cos 2x = cos^2x-sin^2x = 2cos^2x -1#

so:

#int cos^2(2x)sinx = int (2cos^2x -1)^2sinxdx#

Substitute:

#t= cosx#

#dt = -sinxdx#

to have:

#int cos^2(2x)sinx = -int (2t^2-1)^2dt#

#int cos^2(2x)sinx = -int (4t^4 -4t^2+1)dt#

#int cos^2(2x)sinx = -4/5t^5 +4/3t^3-t+C#

and undoing the substitution:

#int cos^2(2x)sinx = - (12cos^5x-20cos^3x+15cosx)/15+C#