Question

求不定积分，详细过程

高等数学 大学数学

Answer 1

方法如下，请作参考：

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Answer 2

∫ x/(1+cosx) dx
=(1/2)∫ x/[cos(x/2)]^2 dx
=(1/2)∫ x[sec(x/2)]^2 dx
=∫ xdtan(x/2)
=xtan(x/2) -∫ tan(x/2) dx
=xtan(x/2) -∫ [sin(x/2)/cos(x/2)] dx
=xtan(x/2) +2∫ dcos(x/2)/cos(x/2)
=xtan(x/2) +2ln|cos(x/2)| + C

Answer 3

∫xdx/(1+cosx) = ∫xdx/{2[cos(x/2)]^2} = ∫xd(x/2)/[cos(x/2)]^2
= ∫xd[tan(x/2)] = xtan(x/2) - ∫tan(x/2)dx = xtan(x/2) - 2∫tan(x/2)d(x/2)
= xtan(x/2) + 2ln|cos(x/2)| + C