求解∫(x+2sinxcosx)／(1+cos2x) dx

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/26 12:20:53

∫ (x + 2sinxcosx)/(1 + cos2x) dx
= ∫ x/(1 + cos2x) dx + ∫ 2sinxcosx/(1 + cos2x) dx
= ∫ x/(1 + 2cos²x - 1) dx + ∫ sin2x/(1 + cos2x) dx
= (1/2)∫ xsec²x dx - (1/2)∫ d(cos2x)/(1 + cos2x)
= (1/2)∫ x d(tanx) - (1/2)∫ d(1 + cos2x)/(1 + cos2x)
= (1/2)xtanx - (1/2)∫ tanx dx - (1/2)ln(1 + cos2x) + C
= (1/2)xtanx + (1/2)ln(cosx) - (1/2)ln(1 + cos2x) + C
= (1/2)xtanx + (1/2)ln(cosx) - (1/2)ln(1 + 2cos²x - 1) + C
= (1/2)[xtanx - ln(cosx)] + C

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