若f’(cosx+2)=tan2x+3sin2x,且f(0)=8,则f(x)=_______.

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问题 若f’(cosx+2)=tan2x+3sin2x,且f(0)=8,则f(x)=_______.

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答案[*]

解析 令t=cosx+2cosx=t-2, cos2x=(t-2)2
    sin2x=1-(t-2)2,tan2x=(t-2)-2-1
f’(t)=(t-2)-2-1+3-3(t-2)2=2+(t-2)-2-3(t-2)2
f(t)=2t-(t-2)-1-(t-2)3+C
f(x)=2x--(x-2)3+C
由   f(0)=+8+C=8C=
因此  f(x)=2x--(x-2)3
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