Chapter 5 Integration

Chapter 5

Integration

Indefinite Integral

or

Antiderivative

1

, 11

nn xx dx c n

n

3 4 24 2x x dx x x c

Find the Particular Solution

or

Solve the Differential Equation

2

4 1, (1,5)

2 4

dyx at

dx

y x x

1. Change into a differential equation.2. Integrate both sides of the equation.3. Find c by plugging in the coordinate.

4. Replace c and write the particular solution.

1st Fundamental Theorem of Calculus

Definite Integral

or

Area Under the Curve on the interval [a,b]

Area below the x-axis is NEGATIVE

( ) ( ) ( )b

af x dx F b F a

3 2 2

14 2 (3) 2(3) (1) 2(1) 4x dx

Approximate the Area Under a Curve

Using a Left-Sided Sum

4 1(1) (2) (3)

3A f f f

2( ) 6 on 1,4 with 3subintervalsf x x x

Approximate the Area Under a Curve

Using a Right-Sided Sum

2( ) 6 on 1,4 with 3subintervalsf x x x

4 1(2) (3) (4)

3A f f f

Approximate the Area Under a Curve

Using a Midpoint Sum

Midpoint Sum

4 1(1.5) (2.5) (3.5)

3A f f f

2( ) 6 on 1,4 with 3subintervalsf x x x

Approximate the Area Under a Curve

Using a Trapezoid Sum

4 1 1(1) 2 (2) 2 (3) (4)

3 2A f f f f

2( ) 6 on 1,4 with 3subintervalsf x x x

Mean Value Theorem (MVT)

or

Average Value

1( ) ( )

b

af c f x dx

b a

( )( ) ( )b

af c b a f x dx Mean Value Theorem

Average Value

Find the x value where you get the

Average Value

1. Find the Average Value.2. Set the original function equal

to the Average Value.3. Solve for x.

4

2

( ) 2 1, 2,4

12 1 2 1

4 2

f x x

x x dx

2nd Fundamental Theorem of Calculus

( ) ( ) 'u

a

df t dt f u u

du

34 22 3 2 8

14 12 192

xdt dt x x x

dx

( )a

af x dx

0

( )a

bf x dx

( )b

af x dx

( )b

af x k dx

( )b b

a af x dx k dx

Integrate an Even Function

0( ) 2 ( )

a a

af x dx f x dx

Integrate an Odd Function

( ) 0a

af x dx

U-Substitution

or

Change of Variables

( ( )) '( ) ( ) ( )f g x g x dx f u du F u C

32 3

2

2 3 ( )

3 2

x x dx u du

u x and du x dx

Find Definite Integral Using

U-Substitution

or

Change of Variables

( )

( )( ( )) '( ) ( )

b g b

a g af g x g x dx f u du

2 (2) 132 3 3

0 (0) 3

2

2 3 ( ) ( )

3 2

g

gx x dx u du u du

u x and du x dx