文档介绍：calculus§()()bbaakfxdxkfxdx???性质:常数因子可以提到积分号外面,即若k为常数1()()()bbbaaafxdxftdtfudu?????性质:定积分的值与积分变量的记号无关,即acalculus[()()]()()bbbaaafxgxdxfxdxgxdx??????性质3:两个函数代数和的定积分等于它们的定积分的代数和,即()()baabfxdxfxdx????性质4:交换定积分的上下限,定积分的值变号,即calculus“性质3的证明”:由定积分的定义,0101()lim(),()lim(),nbiiaxinbiiaxifxdxfxgxdxgx????????????????所以0011()()lim()lim()nnbbiiiiaaxxiifxdxgxdxfxgx?????????????????????01lim()()()()nbiiiaxifgxfxgxdx????????????calculus()()()bcbaacfxdxfxdxfxdx?????性质5:(定积分的区间可加性)对于任意点c,均有[,]()1,1bbaaabfxdxdxba??????性质6:如果在区间上,则2A3Axyab)x(fy?1Acdcalculus补充:不论的相对位置如何,,,例若,cba???cadxxf)(????cbbadxxfdxxf)()(.)()(????bccadxxfdxxf(定积分对于积分区间具有可加性)?badxxf)(????cbcadxxfdxxf)()(则calculus证,0)(?xf?,0)(???if),,2,1(ni??,0??ix?,0)(1??????iinixf},,,max{21nxxx??????iinixf?????)(lim10??.0)(???badxxf则0)(??dxxfba.)(ba?性质7如果在区间],[ba上0)(?xf,calculus证),()(xgxf??,0)()(???xfxg,0)]()([????dxxfxgba,0)()(????babadxxfdxxg于是dxxfba?)(dxxgba??)(.性质7的推论:则dxxfba?)(dxxgba??)(.)(ba?如果在区间],[ba上)()(xgxf?,(1)calculus证,)()()(xfxfxf????,)()()(dxxfdxxfdxxfbababa???????即dxxfba?)(dxxfba??)(.说明:可积性是显然的.|)(xf|在区间],[ba上的)(ba?性质5的推论:dxxfba?)(dxxfba??)(.(2)calculus,()[,]()()()baMmfxabmbafxdxMba?????性质8:(估值定理)设分别是函数在区间上的最大值和最小值,则calculus,x)x(f12??2,m???)(142??4121dx)x(?651?)(1417????4121dx)x(:???在[1,4]上的最小值、最大值分别为:??4121dx)x(所以