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