Post on 16-Jan-2016
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Ch5 Indefinite Integral
Calculus
calculusintegral
calculusaldifferenti
5.1 Antiderivatives and indefinite integral
• Concepts of antiderivatives and indefinite integral• Brief table of indefinite integrals• The property of indefinite integral
Eg. xx cossin
)0(1
ln xx
x
xln is an antiderivatives of x1on ),0(.
Concepts of antiderivatives and indefinite integral
Def: A function F is called an antiderivative of f on an interval I if for all x in I.)()( xfxF
Sinx is an antiderivatives of cosx.
Eg. Does the sign function
0,1
0,0
0,1
sgn)(
x
x
x
xxf
Exists its antiderivative on ？why ？
),(
solution It does exist.
0,
0,
0,
)(
xCx
xC
xCx
xF
Tips: Every function that has jump or removable discontinuity does have its antiderivative.
Suppose there is an antiderivative F(x)
But F(x) isn’t differential at x = 0, therefore, there is no antiderivative.
Questions:
(1) Is there only one antiderivative?
Eg.
xx cossin xCx cossin
（ is constant ）
C
(2) If not, is there any relations?
Tips ：（ 1 ） if ， for any constant ，)()( xfxF C
CxF )( is the antiderivatives of )(xf .
then CxGxF )()( （ is constant ）
CSolution
)()()()( xGxFxGxF
0)()( xfxf
CxGxF )()( （ is constant ）
C
（ 2 ） If F(x) and G(x) are the antiderivatives of f(x)
Con
stant of
integration
Integral sign
integran
d
Definition ：
CxFdxxf )()(被积表达式
Variab
le of in
tegration
denotes dxxf )( .
The family of all antiderivatives of f on the interval I is called the indefinite integral of I
Eg.1 Evaluate
.5dxx
Sol. ,6
56
xx
.
6
65 C
xdxx
solution
Eva. .1
12
dx
x
,1
1arctan 2x
x
.arctan1
12
Cxdx
x
Eg.2
Eg.3 if a curve passes （ 1 ， 2 ）， and the tangent slope is always twice of point of tangency’ s horizontal coordinate ， find the curve’s equation.Solution Suppose the equation of the
curve is ),(xfy
Hence, ,2xdxdy
i.e. )(xf is an antiderivative of x2
,2 2 Cxxdx ,)( 2 Cxxf
And the curve passes （ 1 ， 2 ） ,1 C
Therefore, the equation is .12 xy
According to the definition of indefinite integral ，we know
),()( xfdxxfdxd
,)(])([ dxxfdxxfd ,)()( CxFdxxF
.)()( CxFxdF
Tips ：The operations of Differential and Indefinite Integral are mutually inverse mutually inverse .
example
x
x
1
1
.1
1
Cx
dxx
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.
)1(
Brief table of indefinite integrals
基本积分表
kCkxkdx ()1( is const.);
);1(1
)2(1
Cx
dxx
;ln)3( Cxx
dx
Tips ： ,0x ,ln Cxx
dx
])[ln(,0 xx ,1
)(1
xx
x
,)ln( Cxx
dx,||ln Cx
xdx
denotes .ln Cxx
dx
dx
x211
)4( ;arctan Cx
dx
x21
1)5( ;arcsin Cx
xdxcos)6( ;sin Cx
xdxsin)7( ;cos Cx
xdx
2cos)8( xdx2sec ;tan Cx
xdx
2sin)9( xdx2csc ;cot Cx
xdxx tansec)10( ;sec Cx
xdxxcotcsc)11( ;csc Cx
dxe x)12( ;Ce x
dxa x)13( ;ln
Ca
a x
xdxsinh)14( ;cosh Cx
xdxcosh)15( ;sinh Cx
Eg.4 find .2 dxxx
solution
dxxx 2 dxx 25
Cx
125
125
.72 2
7
Cx
Using formula （ 2 ） Cx
dxx
1
1
dxxgxf )]()([)1( ;)()( dxxgdxxf
Sol. dxxgdxxf )()(
dxxgdxxf )()( ).()( xgxf
We have proved (1).
（ This is true when it is the sum of finite functions ）
Properties
dxxkf )()2( .)( dxxfk
（k is constanst， )0k
Eg.5 Evaluate
Sol.
.)1
2
1
3(
22 dxxx
dxxx
)1
21
3(
22
dxx
dxx
22 1
12
11
3
xarctan3 xarcsin2 C
Eg.6 Evaluate
Sol.
.)1(
12
2
dxxx
xx
dxxxxx
)1(1
2
2
dxxxxx
)1()1(
2
2
dxxx
11
12 dx
xdx
x
1
11
2
.lnarctan Cxx
Eg.7 Evaluate
Sol.
.)1(
2122
2
dxxx
x
dxxx
x
)1(
2122
2
dxxxxx
)1(
122
22
dxx
dxx
22 111
.arctan1
Cxx
Eg.8 Evaluate
Sol.
.2cos1
1
dxx
dx
x2cos11
dx
x 1cos211
2
dxx2cos
121
.tan21
Cx
Tips ： First change the form of the integrand ，then apply the formula in brief table of indefinite integrals.
Eg.9 a curve )(xfy has the tangent slope
xx sinsec2 at the point ))(,( xfx ，and the
intersection with y axis is )5,0( ，find the equation of
the curve.
Solution ,sinsec2 xxdxdy
dxxxy sinsec2
,costan Cxx
,5)0( y ,6 C
The equation of the curve is .6costan xxy
Eg. 10 if the marginal cost of producing x items is 1.92-0.002x and if the cost of producing one item is ￥ 562, find the cost function and the cost of producing 100 items.
Solution let f (x) be the cost function, Then .002.092.1)( xxf
.001.092.1)()( 2 Cxxdxxfxf081.560001.092.1)1(562 CCf
081.560001.092.1)( 2 xxxf
081.742081.56010192)100( f
So, the cost of producing 100 items is ￥ 742.081
Brief table of indefinite integral 5.1
Properties for indefinite integral
def. of antiderivative ： )()( xfxF
Def. of indefinite integral ： CxFdxxf )()(
Mutual inverse relationship
Conclusion