Lesson 1

1. An introduction to
Integration
The inverse process of
Differentiation
IB SL/HL

2. The inverse of differentiation
f (x) = 4x3
+ 6x2
Notation for integration.
Integration is the opposite of
differentiation.
4x3
+ 6x2
( )dx∫
Integrate with respect to x.
4x3
+ 6x2
( )dx∫ =
4x4
4
+
6x3
3
= x4
+ 2x3
Add one to the power and divide
by the new power.
Now differentiate and we should
return to f(x).
Also if we take a function,
differentiate it and then integrate
it we will return to the original
function.
For each of the functions below
differentiate and then integrate
with respect to x.
1. f (x) = 3x2
− 8x + 5
2. g(x) =10x4
− 3x2
+ 4x −10
3. h(x) =
4
x
+12x3
− y
What conclusions can you draw?

3. Integration
This is called indefinite integration.
The formula for this is:
xn
dx =
xn+1
n +1
+ C, n ≠ −1∫

4. Integration questions
Questions Answers

5. Integration questions
Questions Answers
sin x( )dx∫ = −cosx + c
4x2
+ 5x −1( )dx∫ =
4x3
3
+
5x2
2
− x + c
cos x( )dx∫ = sin x + c
ex
( )dx∫ = ex
+ c

6. Multiples
6x3
− 3x2
+12x + 9( )dx∫
can be re-written as,
now it becomes easier to
integrate,
3 2x3
− x2
+ 4x + 3( )dx =∫
3 2x3
− x2
+ 4x + 3( )dx∫
Questions
1. −3sin x( )dx∫
3. 10x2
− 20x3
+ 25x4
( )dx∫
3cosx + c
4ln x + c
10x3
3
− 5x4
+ 5x5
+ c

7. Finding the c value
y = x3
+ 2x2
− 4x + c
(1,4) 4 = (1)3
+ 2(1)2
− 4(1) + c
c = 5
y = x3
+ 2x2
− 4x + 5
Questions
Find f(x) for each of the following.

8. Finding the c value
y = x3
+ 2x2
− 4x + c
(1,4) 4 = (1)3
+ 2(1)2
− 4(1) + c
c = 5
y = x3
+ 2x2
− 4x + 5
Questions
Find f(x) for each of the following.
f (x) =
x2
2
+ 3x −1
f (x) = 3cos x + 2
f (x) = 5ln x + x − 3