Review 5.1-5.3

Calculus

(Make sure you study RS and WS 5.3)

Given f ’(x), find f(x)

f’(x)

f’(x)

Given f ’(x), find f(x)

f’(x)

Basic Integration Rules

CkxkdxRule 1: (k, a constant)

Example 2:

Cx2dx2

Example 3:

Cxdx 22

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 Rule C1n

xx

1nn

n 1

Example 4: Find the indefinite integral dtt 3

Solution: C4t

dtt4

3

Example 5: Find the indefinite integral dxx23

CX52

C

25X

C1

23x

dxx 2

52

51

2

3

2

3

Solution:

Basic Integration Rules

Example 6: Find the indefinite integral dxx13

Solution:

C2x

1C

2x

C13

xxdx

x

12

2133

3

Example 7: Find the indefinite integral dx1Cxdx1 Solution:

Example 8: Find the indefinite integral dxx3 2

Solution: C

x3

C1

3xC

123x

x3dx3x112

22

Here are more examples of Rule 1 and Rule 2.

Evaluate Let u = x2 + 1

du = 2x dx

Multiplying and dividing by a constant

Let u = x2 + 1du = 2x dx

Let u = 2x - 1

du = 2dx

Substitution and the General Power RuleWhat would you let u = in the following examples?

u = 3x - 1

u = x2 + x

u = x3 - 2

u = 1 – 2x2

u = cos x

Example 5a. Find dx13x

x2

Solution: Pick u.

Substitute and integrate:

Example 2a. Find dx13x5

Solution: What did you pick for u?

u = 3x + 1du = 3 dx

Substitute: You must change all variables to u.

Just like with derivatives, we do a rewrite on the square root.

C13x910

Cu32

35

duu35

3du

u5dx13x5 2

32

3

2

1

Example 3a. Find dx1xx 2

332

Solution: Pick u.

Substitute, simplify and integrate:

Find the indefinite integral:1.) 2.)

dxdu

dxdu

xu

1

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 1000.

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