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∫(1+sinx)/(sin3x+sinx)dx求解,
题目详情
∫(1+sinx)/(sin3x+sinx)dx求解,
▼优质解答
答案和解析
解= (1 + sinx)/(sin3x + sinx) dx
= ∫ (1 + sinx)/[2sin(3x + x)/2 * cos(3x - x)/2]
= (1/2)∫ dx/(sin2xcosx) + (1/2)∫ sinx/(sin2xcosx) dx
= (1/2)∫ dx/(2sinxcos²x) + (1/2)∫ sinx/(2sinxcos²x) dx
= (1/4)∫ cscxsec²x dx + (1/4)∫ sec²x dx
= (1/4)∫ cscx(1 + tan²x) dx + (1/4)tanx
= (1/4)∫ cscx dx + (1/4)∫ secxtanx dx + (1/4)tanx
= (1/4)ln| cscx - cotx | + (1/4)(secx + tanx) + C
= ∫ (1 + sinx)/[2sin(3x + x)/2 * cos(3x - x)/2]
= (1/2)∫ dx/(sin2xcosx) + (1/2)∫ sinx/(sin2xcosx) dx
= (1/2)∫ dx/(2sinxcos²x) + (1/2)∫ sinx/(2sinxcos²x) dx
= (1/4)∫ cscxsec²x dx + (1/4)∫ sec²x dx
= (1/4)∫ cscx(1 + tan²x) dx + (1/4)tanx
= (1/4)∫ cscx dx + (1/4)∫ secxtanx dx + (1/4)tanx
= (1/4)ln| cscx - cotx | + (1/4)(secx + tanx) + C
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