1. Calculus (Make sure you study RS and WS 5.3)
Review
Calculus
(Make sure you study RS and WS 5.3)

2. Given f ’(x), find f(x)
f’(x)
f’(x)

3. Given f ’(x), find f(x)
f’(x)

4. Example 3: Basic Integration Rules Rule 1: (k, a constant)
Keep in mind that integration is the reverse of differentiation.
What function has a derivative k?
kx + C, where C is any constant.
Another way to check the rule is to differentiate the result and see
if it matches the integrand. Let’s practice.
Example 2:
Example 3:

5. Example 5: Find the indefinite integral
Basic Integration Rules
Rule 2: The Power Rule n
Example 4: Find the indefinite integral
Solution:
Example 5: Find the indefinite integral
Solution:

6. Example 7: Find the indefinite integral
Here are more examples of Rule 1 and Rule 2.
Example 6: Find the indefinite integral
Solution:
Example 7: Find the indefinite integral
Solution:
Example 8: Find the indefinite integral
Solution:

7. Evaluate
Let u = x2 + 1
du = 2x dx

8. Multiplying and dividing by a constant
Let u = x2 + 1
du = 2x dx
Let u = 2x - 1
du = 2dx

9. u = 3x - 1 u = x2 + x u = x3 - 2 u = 1 – 2x2 u = cos x
Substitution and the General Power Rule
What would you let u = in the following examples?
u = 3x - 1
u = x2 + x
u = x3 - 2
u = 1 – 2x2
u = cos x

10. Example 5a. Find
Solution: Pick u.
Substitute and integrate:

11. Example 2a. Find
Solution: What did you pick for u?
u = 3x + 1
du = 3 dx
You must change all variables to u.
Substitute:
Just like with derivatives, we do a rewrite on the square root.

12. Example 3a. Find
Solution: Pick u.
Substitute, simplify and integrate:

13. Find the indefinite integral:
1.) )

14. Use the log rule to find the indefinite integral
1.) )

15. Find the indefinite integral:
1.) )

16. Find the indefinite integral:
1.) ) x
+ 3 8

17. A population of bacteria is growing a rate of where t is the time in days. When t = 0, the population is 1000.
A.) Write an equation that models the population P in terms of t.
When t = 0, P(t) = 1000,
therefore C = 1000
B.) What is the population after 3 days?
About 7,715
C.) After how many days will the population be 12,000?
6 Days