The Indefinite Integral

Objective: Develop some fundamental results about

antidifferentiation

Definition

• Definition 5.2.1• A function F is called an antiderivative of a function f

on a given interval I if F/(x) = f(x) for all x in the interval.

Definition

• Definition 5.2.1• A function F is called an antiderivative of a function f

on a given interval I if F/(x) = f(x) for all x in the interval.

• For example, the function is an antiderivative of on the interval

because for each x in this interval

2)( xxf

331)( xxF

),(

)()( 2331/ xfxx

dx

dxF

Definition

• However, is not the only antiderivative of f on this interval. If we add any constant C to

this would also be an antiderivative. We will express this as

331)( xxF

331 x

CxxF 331)(

Theorem 5.2.2

• If F(x) is any antiderivative of f(x) on an interval I, then for any constant C the function F(x) + C is also an antiderivative on that interval. Moreover, each antiderivative of f(x) on the interval I can be expressed in the form F(x) + C by choosing the constant C appropriately.

The Indefinite Integral

• The process of finding antiderivatives is called antidifferentiation or integration. Thus, if

then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written

)()]([ xfxFdx

d

CxFdxxf )()(

The Indefinite Integral

• The process of finding antiderivatives is called antidifferentiation or integration. Thus, if

then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written

• Note that if we differentiate an antiderivative of f(x), we obtain f(x) again. Thus

)()]([ xfxFdx

d

)()( xfdxxfdx

d

CxFdxxf )()(

The Indefinite Integral

• The differential symbol dx, in the differentiation and antidifferentiation operations

serves to identify the independent variable. If an independent variable other than x is used, then we would change the notation appropriately.

)()( tftFdt

d

][dx

d

CtFdttf )()(

dx][

The Power Rule in Reverse

• Lets look at the Power Rule again. We are going to do everything in reverse.

• Differentiation Integration Mult. by the exponent Add 1 to the exponent Subt. 1 from exponent Div by the new exponent

The Power Rule in Reverse

• Lets look at the Power Rule again. We are going to do everything in reverse.

• Differentiation Integration Mult. by the exponent Add 1 to the exponent Subt. 1 from exponent Div by the new exponent

2133 33][ xxxdx

d C

xdxx

13

133

The Power Rule in Reverse

• Here are some more examples. This process needs to be memorized.

Cxx

dxxdxx 2/332

21

12/1

1

21

Cx

xdxxdx

x 4

155

5 4

1

15

1

Integration formulas

• Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 324.

Integration formulas

• Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 324.

Properties of the Indefinite Integral

• Theorem 5.2.3• Suppose that F(x) and G(x) are antiderivatives of f(x)

and g(x) respectively, and that c is a constant. Then:

a) A constant factor can be moved through an integral sign.

CxcFdxxfcdxxcf )()()(

Properties of the Indefinite Integral

• Theorem 5.2.3• Suppose that F(x) and G(x) are antiderivatives of f(x)

and g(x) respectively, and that c is a constant. Then:

b) An antiderivative of a sum is the sum of the antiderivative.

CxGxFdxxgdxxfdxxgxf )()()()()]()([

Properties of the Indefinite Integral

• Theorem 5.2.3• Suppose that F(x) and G(x) are antiderivatives of f(x)

and g(x) respectively, and that c is a constant. Then:

b) An antiderivative of a difference is the difference of the antiderivative.

CxGxFdxxgdxxfdxxgxf )()()()()]()([

Properties of the Indefinite Integral

• Theorem 5.2.3 can be summarized by the following formulas.

dxxgdxxfdxxgxf )()()]()([

dxxfcxcf )()(

dxxgdxxfdxxgxf )()()]()([

Example 2

• Evaluate xdxcos4

Example 2

• Evaluate xdxcos4

Cxxdxxdx sin4cos4cos4

Example 2

• Evaluate dxxx )( 2

Example 2

• Evaluate dxxx )( 2

Cxx

dxxxdxdxxx 32)(

3222

Example 3

• Evaluate dxxxx )1723( 26

Example 3

• Evaluate dxxxx )1723( 26

dxxdxdxxdxxdxxxx 1723)1723( 2626

Cxxxx

2

7

3

2

7

3 237

Example 4

• Evaluate dxx2sin

cos

Example 4

• Evaluate dxx2sin

cos

Cxxdxxdxx

x

xdxx

csccotcscsin

cos

sin

1

sin

cos2

Example 4

• Evaluate

dtt

tt4

42 2

Example 4

• Evaluate

dtt

tt4

42 2

Cttdttdtt

dtt

tt

2)2(212 1224

42

Example 4

• Evaluate dx

x

x

12

2

1

111

222

x

xx

Example 4

• Evaluate dx

x

x

12

2

1

111

222

x

xx

Cxxdxx

1

2tan

1

11

Integral Curves

• Graphs of antiderivatives of a function are called integral curves of f. For example, is one integral curve for . All other integral curves have equations of the form

2)( xxf

3

3

1xy

Cxy 3

3

1

Example 5

• Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

Example 5

• Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

• “The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

Example 5

• Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

• “The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

Cxdxx 32

3

1

Example 5

• Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

• Now we use the point (2, 1) to solve for C.

Cxy 3

3

1C 3)2(

3

11

C3

5

3

81

Example 5

• Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

• Now we use the point (2, 1) to solve for C.

Cxy 3

3

1C 3)2(

3

11

C3

5

3

81 3

5

3

1 3 xy

Homework

• Section 5.2

• 1-35 odd