作业帮定积分,
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定积分,
定积分,
定积分,
设M=∫【0,π/2】lnsinxdx
令x=2t.
则M=2∫【0,π/4】lnsin2tdt=2∫【0,π/4】
ln(2sintcost)dt =2∫【0,π/4】ln2dt+2∫【0,π/4】lnsintdt+2∫【0,π/4】lncostdt
而对于N=∫【0,π/4】lncostdt,
令t=π/2-u. 则有N=∫【π/2,π/4】lnsin(π/2-u)(-du)
=∫【π/4,π/2】lncosudu =∫【π/4,π/2】lncostdt
∴M=2∫【0,π/4】ln2dt+2∫【0,π/4】lncostdt+2∫【π/4,π/2】lncostdt
=(πln2)/2+2∫【0,π/2】lncostdt=(πln2)/2+2∫【0,π/2】lnsintdt=(πln2)/2+2M
∴M=(-πln2)/2.