1. Integral Rules; Integration by Substitution

2. Integral Rules; Integration by Substitution
Rules for indefinite integration
1. Constant Multiple Rule :
(Note k is outside of the function)
2. Rule for Negatives :
(Rule 1 with k = 1)
3. Sum and Difference Rule :

3. Integral Rules; Integration by Substitution
The Power and Chain Rule in Integral Form
When u is a differentiable function of x, and n is a rational number (n  1) , the Chain Rule tells us that
This same equation , says that u n+1/(n+1) is one of the antiderivatives of the function u n(d u/ dx) . Therefore ,

4. Integral Rules; Integration by Substitution
From the previous slide:
The integral on the left-hand side of this equation is usually written in the simpler “differential” form ,
obtained by treating the dx’s as differentials that cancel.
To summarize, If u is any differentiable function , then
Remember that we are thinking of the u’s as functions of x

5. Example 1:
One of the clues that we look for is if we can find a function and its derivative in the integrand.
The derivative of is
Note that this only worked because of the 2x in the original.
Many integrals can not be done by substitution.

6. If we let u = g(x) then du = g’(x)dx
Substitution : Running the Chain Rule backward
If we let u = g(x) then du = g’(x)dx
So the integral becomes
Since the F(u) is the antiderivative of f(u)
Then we get this when we substitute
g(x) back in
Notice the use of the small “f” and the big “F”
The method works because F(g(x)) is an antiderivative of
f(g(x)) • g′(x) whenever F is an antiderivative of f :

7. The main idea of integration using substitution
In each case in the previous example and in the examples that follow, we can use integration by substitution only because we have a composite function which has a function on the inside for which we have a derivative on the outside.

8. Example 2:
We solve for because we can see it in the integrand.

9. Example 3: This one is tricky because there does not
seem to be anything on the outside, but since
the derivative of the inside is a constant, it
does not matter .
Solve for dx.

10. Example 4:

11. Remember that sin 4 (x) means (sin x)4
Example 5:
Remember that sin 4 (x) means (sin x)4

12. EXAMPLE 6
Evaluate.

13. SOLUTION

14. EXAMPLE 7
Evaluate.

15. SOLUTION

16. EXAMPLE 8
Evaluate.

17. SOLUTION

18. EXAMPLE 9
Evaluate.

19. SOLUTION…

20. …SOLUTION

21. EXAMPLE 10
Evaluate.

22. SOLUTION

23. Taking the derivative !!! Remember that we can
always check any integration
problem by ????
Taking the derivative !!!