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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
