1. Integration To integrate ex
To be able to integrate functions where the numerator is the derivative of the denominator

2. Integration Remember Differentiation and Integration are inverses
Multiply by power and reduce power by 1
Add 1 to power and divide by new power

3. Chain rule: u = ax du/dx = aex
Differentiation
if f(x) = kex
f `(x) = kex
g`(x) = aeax
if g(x) = eax
Chain rule: u = ax du/dx = a
Chain Rule
y = eu dy/du = eu
dy/dx = eu x a = a eu = aeax
Integration

4. Integration using substitution
The Chain rule for differentiation provides a useful technique for integration
In the chain rule we introduce a new variable u. We can do the same in integration
We must also replace dx
Integrate new function then substitute back.
Chain Rule

5. Integration using substitution Example
Let u = 5x then = 5
To replace dx we can separate the variable to give du = 5 dx which gives
We now have
Which we can now integrate

6. Integration using substitution Example
Integrate
Now substitute back

7. Integration using substitution - special case
Integrate functions where the numerator is the derivative of the denominator ie

8. Integration of Examples
If you want to integrate a function such as or
(note that the numerator is the
denominator differentiated)

9. Integration of Examples
If you want to integrate a function such as
When you do the substitution u = x2 + 1
du/dx = 2x
du/2x = dx
You end up with ….

10. Integration of Examples
If you want to integrate a function such as
When you do the substitution u = x3 - 1
du/dx = 3x2
dx = du/3x2
You end up with ….

11. Integration Examples continued
General
Case

12. Integration Examples continued