Question

高数不定积分问题！

∫sinxcosxdx/sinx+cosx.

Answer 1

∫ (sinxcosx)/(sinx + cosx) dx
= (1/2)∫ (1 + 2sinxcosx - 1)/(sinx + cosx) dx
= (1/2)∫ (sin²x + 2sinxcosx + cos²x - 1)/(sinx + cosx) dx
= (1/2)∫ [(sinx + cosx)² - 1]/(sinx + cosx) dx
= (1/2)∫ (sinx + cosx) dx - (1/2)∫ dx/(sinx + cosx)
= (1/2)(sinx - cosx) - (1/2)∫ dx/[√2•sin(x + π/4)]
= (1/2)(sinx - cosx) - (1/2)(1/√2)∫ csc(x + π/4) dx
= (1/2)(sinx - cosx) - (√2/4)ln|csc(x + π/4) - cot(x + π/4)| + C
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另解：
∫ (sinxcosx)/(sinx + cosx) dx
= ∫ [sin²(x + π/4) - 1/2]/[√2•sin(x + π/4)] dx
= (√2/2)∫ sin(x + π/4) dx - (√2/4)∫ csc(x + π/4) dx
= (√2/2)cos(x + π/4) - (√2/4)ln|tan[(x + π/4)/2]| + C
= (1/2)(sinx - cosx) - (√2/4)ln|tan(x/2 + π/8)| + C
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它们的答案表示不同，但是相等的

Answer 2

dx/sinx+cosx这个是一起的吗？还是怎么的啊？用括号分一下行吗？

追问

被积函数是sinxcosx/sinx+cosx。