变限积分的求导公式及其应用

ACADEMYO172A16744810201219005102125()Fx=20()()xxtftdt−∫x22220000()()()()()xxxxFxxftdttftdtxftdttftdt=−=−∫∫∫∫20()()xFxxtfxtdt−−∫()=uxt20()()()xFxxtfxt=−−∫2()xxxdtufudu−=−∫5()0()1(,)2txatxatfddaττξτξτ++−−∫∫1αxβxabfxtD{xtαxtβxaxb}fxtx(,)dfxtdxDLeibniz()()''()()(,)()((),)()((),t)+xxxxddfxtdtxfxtxfxdxdxββααββαα=−∫∫(,)fxtdt1()()()(,),()(),xxFxfxtGxFxdtβα==∫()()()()()()()()xxxxxxGxxGxFxxdtFxdtββαα+Δ+Δ+Δ−=+Δ−∫∫(+)()()()()()xxxxxFxxdtFxxdtβββαΔ=+Δ++Δ∫∫()()xxxαα+Δ+∫()()()()xxFxxdtFxdtβα+Δ−∫(+)(+)()()()()xxxxxxFxxdtFxxdtβαβαΔΔ=+Δ−+Δ∫∫()()xxβα+∫[()()]FxxFxdt+Δ−'0()()()limxGxxGxGxxΔ→+Δ−=Δ()()()()0011lim()lim()xxxxxxxxFxxdtFxxdtxxβαβα+Δ+ΔΔ→Δ→=+Δ−+ΔΔΔ∫∫01limxxΔ→+Δ()()[()()]xxFxxFxdtβα+Δ−∫2ACADEMY()(+)11()()000lim()=lim()=lim()[()()]xxxxxxxxxFxxdtFdtFxxxββββξξββ+ΔΔΔ→Δ→Δ→+Δ+Δ−∫∫1((),())xxxξββ∈+Δ()()'100()()()limlim()()(())xxxxxFxxdtxxxFxFxxxββββξββ+ΔΔ→Δ→+Δ+Δ−==ΔΔ∫3(+)()'200()()()limlim()()(())xxxxxFxxdtxxxFxFxxxααααξααΔΔ→Δ→+Δ+Δ−==ΔΔ∫42((),())xxxξαα∈+Δfxt()()()()00[()()]()()limlimxxxxxxFxxFxdtFxxFxdtxxββααΔ→Δ→+Δ−+Δ−=ΔΔ∫∫()()0()()limxxxFxxFxdtxβαΔ→+Δ−=Δ∫()()()xxdFxdtdxβα=∫5352()'''()0()()()lim()(())()(())xxxGxxGxGxxFxxFxxβαββααΔ→+Δ−==−+Δ∫()dFxdtdx()()''()()(,)()((),)()((),)(,)xxxxddfxtdtxfxtxfxtfxtdtdxdxββααββαα=−+∫∫1()''()()()(())()(())xxdftdtxfxxfxdxβαββαα=−∫60()()xdftdtfxdx=∫7122111431[()]lim(1)xtxxtfududttdt→+∫∫∫ff10122111431[()]lim(1)xtxxtfududttdt→+∫∫∫22114281()lim3(1)21xxxxfudutdtxx→=++∫∫2211421()lim62(1)xxxfudutdt→=+∫∫2214812()lim122(1)21xxxfxtdtxx→−=++∫22141()=lim241xxfxtdt→−+∫'2812()lim481xxfxxx→−=+'(1)242f−=2221()()xtfxxtedt−=−∫fx+242'221()2()2xxtfxxxxexedt−−=−+∫2212xtxedt−=∫fxx01fx101101f10f021101(1)2ttedte−−==−∫1999fx201(2)arctan2xtfxtdtx−=∫f1121()fxdx∫u2xt20(2)(2)()xxxtfxtdtxufudu−=−∫∫221(2)()arctan2xxxufudux−=∫x1242()()1xxxfuduxfxx=++∫。x1213()4fxdx=∫2007fx04πf1ffx()100cossin()cossinfxxttftdttdttt−−=+∫∫66x1'cossin[()]()cossinxxffxfxxxx−−=+''cossincossin(),()cossincossinxxxxxfxxfxxxxx−−==++cossin()(cossin).cossinxxfxdxInxxCxx−==+++∫f00C0fxIncosxsinx122()+41(,)()2xatuxtedatξϕξξπ−−∞−∞=∫2,(,0)()txxuauuxxϕ==22()34.xyatyedy−−+∞−∞∫uxtt22()3+4211(,)()()22xattuxttedaξππϕξξ−−−∞−∞−=∫22211()42xatatξϕπ+∞−∞+−∫22()4()xatedξϕξξ−−22()431()4xatedatξϕξξπ−−+∞−∞=−∫22()24321()()8xatxedattξξϕξξπ−−+∞−∞+−∫1x22()4211(,)()()22xatxuxtxedatatζξϕξξπ−−+∞−∞=−−∫22()4211(,)()22xatxxuxtedatatξϕξξπ−−+∞−∞=−∫22211()()22xatatξϕπ+∞−∞+−∫22()4()xatedξϕξξ−−22()431()4xatedattξϕξξπ−−+∞−∞=−∫2521()()8xattξϕξπ+∞−∞+−∫22()4)xatedξξ−−φxx3uxtx36xta122()3342(,)2(6)xyatyedyatuxtatxxtππ−−+∞−∞==+∫1.M.20092.M.2009