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

Given f ’(x), find f(x) f’(x)

Given f ’(x), find f(x) f’(x)

Basic Integration Rules Rule 1: (k, a constant) Example 2: Example 3: 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.

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

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

Evaluate Let u = x du = 2x dx

Multiplying and dividing by a constant Let u = x du = 2x dx Let u = 2x - 1 du = 2dx

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

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

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

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

Find the indefinite integral: 1.) 2.)

Use the log rule to find the indefinite integral 1.) 2.)

Find the indefinite integral: 1.) 2.)

Find the indefinite integral: 1.) 2.) x+ 3 8

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