Chapter 3 Integral Calculus of One Variable Functions：一元函数积分学3章

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,Chapter 3 Integral Calculus of,One Variable Functions,第三章 一元函数积分学,Eg.,Def.1,1 The,Concept and Properties of Indefinite Integrals,一、,Concept of Antiderivatives and Indefinite Integrals,原函数存在定理(Existence Theorem of Antiderivatives)：,In words,the continuous functions must have antiderivatives,.,Questions:,(1)Is there only one antiderivative?,Eg.,（is a constant,）,(2)If not,is there any relations?,原函数的表示,(,Representation of Antiderivatives,),If F,(,x,),is,an antiderivative,of f,(,x,),on an interval,I,then,G(,x,),is,an antiderivative,of f,(,x,),on an interval,I,if and only if,G(,x,),is,of the form,G(,x,)=,F,(,x,),+C,for all,x,I,Where C,is a constant,.,Proof.,Any Constant,Integral Sign,Integrand,Definition of Indefinite Integral,：,Integrand Expression,Integral Variable,On the interval,I,the antiderivative of f,(,x,),with any,constant,is called the indefinite integral of f,(,x,),on,I,denoted by,Eg.1,Evaluate,Sol.,Sol.,Eg.2,Evaluate,Eg.3,If a curve passes,(,1,2,),and the tangent slope is always twice of point of tangency s horizontal coordinate,find the curves equation.,Sol.,Suppose the equation of the curve is,Hence,And the curve passes,(,1,2,),Therefore,the equation is,According to the definition of,indefinite integral,we know,Tips：,The operations of,Differential,and,Indefinite,Integral,are,mutually inverse,.,Example,Thinking process,Getting the formula of indefinite integrals from the formulas of differential？,Tips,Because the operations between differential and indefinite integral are mutual inverse,we can get the formula of indefinite integrals from the formulas of differential,二、根本积分表(Basic Integration Tables),Basic Integration Tables(1),is a constant,);,Tips：,Eg.4,Evaluate,Sol.,Using formula(2),Proof,We have proved(1).,This is true when it is the sum of finite functions,三、,The,Properties of Indefinite Integrals,Eg.5,Evaluate,Sol.,Sol.,Eg.6,Evaluate,Sol.,Eg.8,Evaluate,Sol.,Eg.7,Evaluate,Eg.9,Evaluate,Sol.,Sol.,Eg.10,Evaluate,Eg.11,Evaluate,Sol.,Tips：,First change the form of the integrand，then apply the formula in,basic integration tables,.,Sol.,The equation of the curve is,1.,不定积分的概念,原函数与不定积分的定义,不定积分的性质,根本积分表,2.,直接积分法:,利用,恒等变形,及 根本积分公式进行积分.,常用恒等变形方法,分项积分,加项减项,利用三角公式,代数公式,积分性质,内容小结,1.,证明,2.假设,提示:,思考与练习,是,的原函数,那么,提示:,3.假设,的导函数为,那么,的一个原函数,是().,提示:,求,即,B,?,?,或由题意,其原函数为,4.假设,提示,:,5.求以下积分:,解：,6.,求不定积分,求,A,B,.,解:,等式两边对,x,求导,得,7.,练习题,练习题答案,