The Integral
• Indefinite Integral………...………….…2
• Integration by Substitution………..…4
• Definite Integral………...7
Indefinite Integral
An antiderivative of the function f is a function F so that
( ) ( ),
F x′ = f x ∀ ∈x I
Example: ( ) 1 3 3
F x = x + C is antiderivative of f (x) = x2
because of . F x′( ) = f x( ) because of . F x( ) = f x( )
The antiderivative of the function f is not unique, but they are different by constant.
The collection of antiderivatives of function f is called the indefinite integral of f with respect to x and denote by
( ) ( )
1
1. , 1
1
2. sin cos
r
r x
x dx C r
r
x dx x C
+
= + ≠ −
+
= − +
Some of Integral
Formulas
2 2
3. cos sin
4. sec tan
5. csc cot
x dx x C
x dx x C
x dx x C
= +
= +
Integration
by Substitution
Let , and F antiderivative of f , then
( ),
( )
u
=
g x
du
=
g x dx
′
( ( )) ( ) ( ) ( ) ( ( ))
f g x g x dx′ = f u du = F u + C = F g x + C
Example: Example:
( )
sin 2x +1 dx
Answer: Let 2 1, 2 1
2 u = x + du = dx → dx = du
(
)
(
)
1
sin 2 1 sin 2
1 1
cos cos 2 1
2 2
x dx u du
u C x C
+ =
= − + = − + +
2. Find
3 2
2 1 1, 3
3
u x du x dx dx du
x
= + = → =
then ( 3 1)10 5 10 5 2 1 10 3
3 3
du
x x dx u x u x du x
+ = =
+ x dx
x3 1)10 5
(
Answer: Let
chapter
6
The Integral
because then , thus u = x3 +1
x
3=
u
−
1
3 10 5 1 10 1 11 10
3 3
12 11
1 1
36 33
3 12 3 11
1 1
36 33
( 1) ( 1)
( 1) ( 1)
x x dx u u du u u du
u u C
x x C
+ = − = −
= − +
1 2
Sigma Notation
b definite integral as limit of Riemann sum.
Definite Integral
b
a x2 xk−1 xk b
4. Defined Riemann sum
=
The Integral
k
Use Riemann sum to evaluate
2
0
(x−2)dx
Answer:
(i) Divide interval [0,2] into n subinterval all of the same length, that is
Example
n
x = 2
∆ x0 = 0
n x x1 = 0+∆ = 2
n .
x
x2 = 0+2∆ = 22
n i
i i x
x = 0+ ∆ = 2
0 2
x
∆ ∆x
x
∆
x
∆
1
(ii) Choose ci = xi
(iii) Defined Riemann sum
( )
(
)
(
2)
2 2 4 4
2
1 1 1 1 1
4 4
2 1
n n n n n
i i
i i n n n n
i i i i i
f c x i
n n
= = = = =
∆ = − = − = −
chapter
6
The Integral
1 1 1 1 1
2
4 ( 1) 4 2
2 2
i i i n i n i
n n
n
n n n
= = = = =
+
= − = − +
(iv) If thenn →∞
(
)
2
2
0
( 2) lim 2 n 2
n
x dx
→∞
Remark: If function y = f(x) positive on interval [a,b] then definite integral = area of region under the graph of y = f(x) over the interval [a,b].
Some properties of definite integral
chapter
6
The Integral
[
p f x q g x dx]
p f x dx q g x dxa b
a b
a b
( ) + ( ) = ( ) + ( ) 1. Linear
2. If a < b < c, then
f x dx f x dx f x dx
a c
a b
b c
f x dx
The Integral
5. If f(x) is an even function, then f x dx f x dx
Example: Evaluate
−
Answer:
4 2 4 2
The Fundamental
Theorem of Calculus Part 1
If f is continuous on [a,b] then is( ) ( )
x
a
F x = f t dt
continuous on [a,b] and differentiable and its
The Fundamental Theorem of Calculus Part 1
continuous on [a,b] and differentiable and its
( ) ( ) ( )
x
a
d
F x f t dt f x
dx
′ = =
( )
( ) ( ( )) ( )
u x
d
f t dt f u x u x
dx = ′
From the fundamental theorem of calculus part 1, we can derive:
chapter
6
The Integral
a
dx
( )
( )
( ) ( ( )) ( ) ( ( )) ( )
v x u x
d
f t dt f v x v x f u x u x
+
The Integral
The Fundamental
Theorem of Calculus Part 2
If f is continuous at every point of [a,b] and F is
any antiderivative of f on [a,b] then
The Fundamental Theorem of Calculus Part 2
any antiderivative of f on [a,b] then
( ) ( ) ( )
b
a
( )
Example
−
The Integral
For problems 1-5, find the antiderivative F(x) + C of f(x).
5 10
3 )
(x = x2 + x+ f
) 6 7
20 ( )
(x = x2 x7 − x5 + f
1.
2.
Problem Set 1
f x
x x
( ) = 1 + 6
3 7
f x x x
x
( ) = 2 − 3 +1
3 2
2
3 4
( )
f x = x−
2.
3.
4.
(
x2 − 4 2)
3 x dx(
x2 − 3x + 2)
2(
2x − 3)
dx6.
7.
For problems 1-5, evaluate each integral given.
Problem Set 2
3x 3x2 + 7 dx
(
5x2 +1 5)
x3+ 3x −2 dx3
2 2 5
y
y
dy +
(
cos42x)
(
− 2 sin 2x dx)
8.
9.
10.
For problem 12 - 15, evaluate .f x dx( )
0 5
f x x x
x x
( ) ,
,
= + ≤ <
− ≤ ≤
2 0 2
6 2 5
x , 0≤ x <1
12.
Problem Set 3
f x
x x
x
x x
( )
, ,
,
=
≤ <
≤ ≤ − < ≤
0 1
1 1 3
4 3 5
13.
14. f(x) = |x -1|
3 1 3
4
2 )
(x x x
f = −
3 2 3 1
For problem 16 - 21, evaluate each integral given.
8 7 2 2
sin cos