1、Concepts and properties of indefinite integrals1Concepts of Indefinite Integrals (Antiderivatives 原函数原函数)A function F(x)is called an anti-derivative of a function f(x)on an interval I if F(x)=f(x)for all points in the interval.2()()Fxf x f(x)is the derivative of function F(x)Example(sin)cosxx 2(sin)
2、cosxx 3(sin)cosxx 6(sin)cosxx()cos()?f xxF x Concepts of Indefinite Integrals (Indefinite Integral 不定积分不定积分)The set of all anti-derivatives of f(x)on the interval I is called the indefinite integral of f(x)with respect to x on I,denoted by3().f x dx Integrand 被积函数被积函数Integration sign 积分号积分号Expressio
3、n under the integral sign被积表达式被积表达式Concepts of Indefinite IntegralsHow to find the set of all anti-derivatives of function f(x)?4()()f x dxF xC Example(sin)cosxx cossinxdxxC Once we have found one anti-derivative F(x)of a function f(x),the other anti-derivatives differ from it by a constant.()()Fxf
4、x 2()2xx 22xdxxC Basic Indefinite Integrals5()()f x dxF xC()()Fxf x kdxkxC (k is a constant)2csccotxdxxC 11xx dxC (1 )sectansecxxdxxC ln|dxxCx csccotsecxxdxxC lnxxaa dxCa (0a ,1a )2arcsin1dxxCx xxe dxeC 2arctan1dxxCx cossinxdxxC sinhcoshxdxxC sincosxdxxC coshsinhxdxxC 2sectanxdxxC Properties of Inde
5、finite Integrals6Property 1()()f x dxf x ()()df x dxf x dx ()()fx dxf xC()()df xf xC,or,or Property 2(Linear property)()()kf x dxkf x dx ()()()()f xg xdxf x dxg x dx (1),k is a constant;(2)Some Examples7Solution 212121112xxxdxdxdxxdxdxxxx 2ln|.xxxCEvaluate 21.xdxx .Evaluate 22.sincosdxxx Solution222
6、222sincossincossincosdxxxdxxxxx tancotxxC22seccscxdxxdxSome Examples8Solution2222224411(1)(1)11111xxxxxxdxdxdxdxxx 3221arctan.31(1)xxxCxdxxdx Evaluate 42.1xdxx .Evaluate 2sin.2xdx Solution211sin(1cos)(1cos)222xdxx dxx dx12(sin).xxC11cos22dxxdxSome Examples9RightWrongChecking Correctness of an Indefinite IntegralReason:The derivative of the right-hand side is the integrand:Reason:The derivative of the right-hand side is not the integrand:(1)cossincosxxdxxxxC(sincos)cossinsin0cosdxxxCxxxxxxdx (2)cossinxxdxxx C (sin)cossin0cosdxxCxxxxxdx
