不定积分试题:这是积分(x+sinx)/(1+cosx)dx

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不定积分试题:这是积分(x+sinx)/(1+cosx)dx

不定积分试题:这是积分(x+sinx)/(1+cosx)dx
不定积分试题:这是积分(x+sinx)/(1+cosx)dx

不定积分试题:这是积分(x+sinx)/(1+cosx)dx
∫(x+sinx)/(1+cosx)dx
=∫xdx/(1+cosx)+∫sinxdx/(1+cosx)
=∫xd(x/2)/[cos(x/2)]^2 +∫tan(x/2)dx
=∫xdtan(x/2)+∫tan(x/2)dx
=xtan(x/2)-∫tan(x/2)dx+∫tan(x/2)dx+C
=xtan(x/2) +C

∫sinx /(1+cosx) dx = ∫tan(x/2) dx
1+cosx = 2 cos²(x/2),[tan(x/2)] ' = (1/2) sec²(x/2)
∫x /(1+cosx) dx = ∫x d[tan(x/2)] = x * tan(x/2) - ∫tan(x/2) dx
原式 = x * tan(x/2) + C