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,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,*,Question,1.Find the limit,2.Evaluate,Example,Ex.,Suppose find f(x).,Sol.,Let then,so,Therefore,Question,Ex.,Find f if it is continuous and,Sol.,Ex.,Find f is it is differentiable on 0,1, f(0)=0 and,Sol.,Indefinite integral,The family of all antiderivatives of f is called the indefinite integral of f and denoted by,If F is an antiderivative of f, then,Table of indefinite integrals (page 406),Ex.,Substitution rule,Ex.,Find the indefinite integral,Sol.,Introducing a new variable then the,differential of,u,is,So,Substitution rule,Suppose F is an antiderivative of f, that is,Then by the chain rule,This is,The Substitution Rule,If is differentiable, then,That is to say, to find the indefinite integral,we can substitute u=g(x), thus and,Example,Ex.,Find the indefinite integrals:,Sol.,(1) Since let,Example,Ex.,Find,Sol.,Example,Ex.,Find,Sol.,Let then and,Example,Ex.,Find,Sol.,Example,Ex.,Sol.,Substitution rule for definite integral,The Substitution Rule,for definite integral: Suppose F is,an antiderivative of f, then,Ex.,Evaluate,Sol.,Example,Ex.,Evaluate,Sol.,Integrals of symmetric functions,Property of symmetry: Suppose f is continuous on -a,a.,(1) If f is even, then,(2) If f is odd, then,Ex.,since the integrand is odd.,Integrals of periodic functions,Suppose f is continuous and periodic, and T0 is a period.,Then,Proof.,let,x,=t+T, then,Example,Ex.,Find the definite integral,Sol.,By substitution,Example,Ex.,If f is continuous and find,Sol.,Let then,Therefore,Homework 14,Section 5.4: 23, 34,Section 5.5: 11, 29, 36, 41, 44, 45,
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