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1、6.2、微積分基本定理設(shè)f(t)在[a,b]上可積,x則對(duì)任意x?[a,b],f(t)在[a,x]上也可積,且定積分?f(t)dt是一個(gè)數(shù).a若固定函數(shù)f(t)和積分下限a,xx則對(duì)于任意一個(gè)x?[a,b],有唯一確定的數(shù)?f(t)dt與之對(duì)應(yīng),記F(x)??f(t)dt.aax因此從函數(shù)的觀點(diǎn)看,定積分F(x)??f(t)dt是其積分上限x的函數(shù),ax且它的定義域?yàn)閇a,b].我們稱(chēng)F(x)??f(t)dt為變上限積分.ab同樣地,G(x)??f(t)dt,x?[a,b]也是關(guān)于x的一個(gè)函數(shù),稱(chēng)其
2、為變下限積分.xx變上限積分F(x)??f(t)dt,x?[a,b];a統(tǒng)稱(chēng)為變限積分.b變下限積分G(x)??f(t)dt,x?[a,b]x變限積分可用于構(gòu)造非初等函數(shù):x2?t例:F(x)??edt,x?[0,1]是一個(gè)非初等函數(shù).0一、變限積分與原函數(shù)x定理6.1:設(shè)f(x)在[a,b]上可積,則F(x)??f(t)dt是[a,b]上的連續(xù)函數(shù).a注:F(x)在[a,b]上連續(xù)指F(x)在(a,b)上處處連續(xù),在x?a點(diǎn)右連續(xù),在x?b點(diǎn)左連續(xù).(1):先證:F(x)在(a,b)上連續(xù).設(shè)x0
3、?(a,b),對(duì)x0的任一改變量?x.x0??xx0x0??x當(dāng)x??x?(a,b)時(shí),有F(x??x)?F(x)?f(t)dt?f(t)dt?f(t)dt.000???aax0由于f(x)在[a,b]上可積,故f(x)在[a,b]有界.設(shè)f(x)?M,x?[a,b],M?0.x0??xx0??xx0??x當(dāng)?x?0時(shí),F(x??x)?F(x)?f(t)dt?f(t)dt?Mdt?M?x;00?x?x?x000x0??xx0當(dāng)?x?0時(shí),F(x??x)?F(x)?f(t)dt?f(t)dt00?x?
4、x??x00x0x0??x??xf(t)dt??x?Mdt?x?M(??x);?F(x0??x)?F(x0)?M?x.00故當(dāng)?x?0時(shí),有F(x??x)?F(x),從而F(x)在x點(diǎn)連續(xù),即F(x)在(a,b)上處處連續(xù).000(2):當(dāng)x?a時(shí),對(duì)x的任一改變量?x?0,00x0??xx0x0??x當(dāng)x??x?[a,b]時(shí),有F(x??x)?F(x)?f(t)dt?f(t)dt?f(t)dt.000???aax0由于f(x)在[a,b]上可積,故f(x)在[a,b]有界.設(shè)f(x)?M,x?[a
5、,b],M?0.x0??xx0??xx0??x?F(x??x)?F(x)?f(t)dt?f(t)dt?Mdt?M?x.00?x?x?x000?故當(dāng)?x?0時(shí),有F(x??x)?F(x),從而F(x)在x?a點(diǎn)右連續(xù).000x定理6.2:設(shè)f(x)在[a,b]上連續(xù),則F(x)??f(t)dt在[a,b]上連續(xù)可導(dǎo),ax且F?(x)?(?f(t)dt)??f(x),x?[a,b].a注:F(x)在[a,b]上可導(dǎo)指F(x)在(a,b)上處處可導(dǎo),在x?a點(diǎn)右可導(dǎo),在x?b點(diǎn)左可導(dǎo).F(x)連續(xù)可導(dǎo)是指
6、:F(x)的導(dǎo)函數(shù)連續(xù).x因此只要證明:F?(x)?(?f(t)dt)??f(x),x?[a,b],即導(dǎo)函數(shù)F?(x)?f(x)是連續(xù)的.a先證:F(x)在(a,b)上可導(dǎo).設(shè)x0?(a,b),對(duì)x0的任一改變量?x使得x0??x?(a,b),F(x??x)?F(x)1x0??xx01x0??x00??那么當(dāng)?x?0時(shí),有????f(t)dt??f(t)dt????f(t)dt.?x?xaa?xx01x0??x1由積分中值定理知:存在?介于x與x??x之間,使得f(t)dt?f(?)?x?f(?).
7、00??xx0?x由于?x?0時(shí),??x,及f(x)在x點(diǎn)連續(xù),00?F(x0??x)?F(x0)?故lim?limf(?)?limf(?)?f(x),即F?(x)?f(x).??000?x?0??x??x?0??x0由x的任意性知:F(x)在(a,b)上處處可導(dǎo),且其導(dǎo)函數(shù)為f(x).0當(dāng)x?a時(shí),對(duì)x的任一改變量?x?0,使得x??x?[a,b],000F(x??x)?F(x)1x0??xx01x0??x00??那么當(dāng)?x?0時(shí),有????f(t)dt??f(t)dt????f(t)dt.?x?
8、xaa?xx01x0??x1由積分中值定理知:存在?介于x與x??x之間,使得f(t)dt?f(?)?x?f(?).00??xx0?x??由于?x?0時(shí),??x,及f(x)在x點(diǎn)連續(xù),故00?F(x0??x)?F(x0)?lim?limf(?)?limf(?)?f(x),即F?(x)?f(x).?????000?x?0??x??x?0??x0故F(x)在x?a點(diǎn)右可導(dǎo),且導(dǎo)數(shù)為f(a).xb設(shè)變上限積分F(x)??f(t)dt,變下限積分G(x)??f(t)dt,x?
