作业帮求∫sinxcosx/(sinx^4+cosx^4 )dx
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/07/05 10:23:29
求∫sinxcosx/(sinx^4+cosx^4 )dx
sinxcosx/[sinx^4+cosx^4]
=1/2sin2x/[(sinx^2+cosx^2)^2-2sinx^2cosx^2]
=1/2sin2x/(1-(sin2x)^2/2)
=sin2x/(2-(sin2x)^2)
=sin2x/(1+(cos2x)^2)
∫[sinxcosx]dx/[sinx^4+cosx^4]
=∫sin2xdx/[1+(cos2x)^2]
=-1/2∫dcos2x/(1+(cos2x)^2)
= -1/2arctan(cos2x)+C
=1/2sin2x/[(sinx^2+cosx^2)^2-2sinx^2cosx^2]
=1/2sin2x/(1-(sin2x)^2/2)
=sin2x/(2-(sin2x)^2)
=sin2x/(1+(cos2x)^2)
∫[sinxcosx]dx/[sinx^4+cosx^4]
=∫sin2xdx/[1+(cos2x)^2]
=-1/2∫dcos2x/(1+(cos2x)^2)
= -1/2arctan(cos2x)+C
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