CALCULUS CHAPTER 6 NOTES

SECTION 6-1 (Day 1) Indefinite Integrals

Indefinite Integrals: If F is the antiderivative of f:

∫ 𝒇(𝒙)𝒅𝒙 = +

- C is called

Some key antiderivatives:

∫ 𝒙𝒏 = ∫𝒅𝒙

𝒙=

∫ 𝒆𝒙 = ∫ 𝒆𝒌𝒙 =

∫ 𝒔𝒊𝒏 𝒙 𝒅𝒙 = ∫ 𝒔𝒊𝒏 𝒌𝒙 𝒅𝒙 =

∫ 𝒄𝒐𝒔 𝒙 𝒅𝒙 = ∫ 𝒄𝒐𝒔 𝒌𝒙 𝒅𝒙 =

∫ 𝒔𝒆𝒄𝟐 𝒙 𝒅𝒙 = ∫ 𝒄𝒔𝒄𝟐 𝒙 𝒅𝒙 =

∫ 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙 𝒅𝒙 = ∫ 𝒄𝒔𝒄 𝒙 𝒄𝒕𝒏 𝒙 𝒅𝒙 =

Key Reminder:

REMEMBER TO BRING ALL CONSTANTS AND NEGATIVES

Examples:

∫(𝟑 𝒄𝒐𝒔 𝒙 − 𝒄𝒐𝒔 𝟑𝒙) 𝒅𝒙 =

∫(𝟏

𝒙 − 𝟐+ 𝒔𝒊𝒏 𝟓𝒙 − 𝒆−𝟐𝒙) 𝒅𝒙 =

∫ 𝒄𝒐𝒔 𝟐 𝒙 𝒅𝒙 =

ASSIGNMENT: Page 312 #3 – 6, 9, 10, 13, 15, 17, 19, 22,

CALCULUS CHAPTER 6 NOTES

SECTION 6-1 (Day 2) Solving Differential Equations

Recall what the differential form is of an equation:

𝒅𝒚

𝒅𝒙= 𝟒𝒙𝟐 − 𝒔𝒊𝒏 𝟐𝒙 +

𝟏

𝒙

Initial Conditions – when a point is given that lies

Example: Solve this differential equation:

𝒅𝒚 = (𝒙−𝟐

𝟑⁄ ) 𝒅𝒙

Given the initial condition: y(-1) = -5, find the original equation.

Example: Given 𝒂 = 𝒔𝒊𝒏 𝜽, find s(t) when v(0) = 0 and s(0) = -3.

ASSIGNMENT: Page 313 #25 - 27, 29, 31 – 34, 36, 41, 42

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CALCULUS CHAPTER 6 NOTES

SECTION 6-1 (Day 3) Slope Fields

SLOPE FIELDS:

Definition: A Slope Field is plot of short line segment with slopes f(x, y) such that:

𝒅𝒚

𝒅𝒙= 𝒇(𝒙, 𝒚)

This is what a slope field looks like.

Sketch the possible solution to slope field given f(0) = 2.

On the axis below, sketch the slope field of the following differential equation:

𝒅𝒚

𝒅𝒙=

𝒙

𝒚

Now, solve the possible differential equation by separation of variables.

ASSIGNMENT: SLOPE FIELDS HANDOUT #1 – 16

CALCULUS CHAPTER 6 NOTES

SECTION 6-2 (Day 1) Substitution Method

Substitution Method -

Examples:

∫(𝟐𝒙 + 𝟑)𝟕 𝒅𝒙 =

∫ 𝟔√𝟑𝒙 − 𝟏 𝒅𝒙 =

∫𝒅𝒙

(𝟓 − 𝒙)𝟑=

∫𝒍𝒏 𝒙

𝒙 𝒅𝒙 =

REMINDERS:

1. All Constants

2. No Variables brought out

3. Never bring any variables (x’s) over to the du

ASSIGNMENT: Page 321 – 322 #2 – 4, 6, 8, 9, 13, 17, 24

CALCULUS CHAPTER 6 NOTES

SECTION 6-2 (Day 2) Substitution w/ Trig Functions

Examples:

∫ 𝟕𝒔𝒊𝒏𝟔 𝒙 𝒄𝒐𝒔 𝒙 𝒅𝒙 =

∫ 𝒕𝒂𝒏𝟑

𝝅𝟒⁄

𝟎

𝒙 𝒔𝒆𝒄𝟐𝒙 𝒅𝒙 =

Making a U-Substitution:

Example:

∫ 𝒄𝒐𝒔−𝟑

𝝅𝟔⁄

𝟎

𝟐𝜽 𝒔𝒊𝒏 𝟐𝜽 𝒅𝜽 =

ASSIGNMENT: Page 321 – 322 #11, 14, 16, 18, 19, 21, 22, 34, 36, 37

CALCULUS CHAPTER 6 NOTES

SECTION 6-2 (Day 3) Separating Variables

Recall Solving a Differential Equation:

𝒅𝒚

𝒅𝒙= (𝒚 + 𝟓)(𝒙 + 𝟐)

1. Separate

2. Integrate

3. Add

4. Solve

5. Find C (If possible)

Solve the differential equation below by separating variables and find C given the initial value given by y(0) = 1.

𝒅𝒚

𝒅𝒙= (𝒄𝒐𝒔 𝒙)𝒆𝒚+𝒔𝒊𝒏 𝒙

ASSIGNMENT: Page 322 #42, 43, 44

CALCULUS CHAPTER 6 NOTES

SECTION 6-3 Integration by Parts

Integration by Parts is derived by integrating the Product Rule.

When to Use:

Evaluate:

∫ 𝒙 𝒄𝒐𝒔 𝒙 𝒅𝒙

Choose: Derivative Antiderivative

Multiple Integration by Parts: (Called

∫ 𝒙𝟐𝒆−𝒙 𝒅𝒙 =

Choose: Derivative Antiderivative

ASSIGNMENT: Page 328 # 2, 15, 16, 19

CALCULUS CHAPTER 6 NOTE

SECTION 6-4 Exponential Growth and Decay

Recall the equation used to calculate an amount compounded continuously:

Substituting: y for A; and y0 for P:

The derivative of this equation with respect to t is:

𝒅𝒚

𝒅𝒕= 𝒌 ∙ 𝒚

So, anytime you see this equation, its antiderivative is:

Also, recall calculating the amount compounded using a fixed rate:

𝑨 = ( +

)

Example: Suppose you deposit $1200 in an account that pays 4% annual interest. How much will you have 6 years later if the interest is:

a.) Compounded Continuously:

b.) Compounded Quarterly:

Radioactive Decay (Half-Life)

The half-life of a certain element is 25 days. If 100 grams of the substance is present

initially, use 𝒚 = 𝒚𝟎 𝒆𝒌𝒕 (where t is measured in days) law of exponential change formula to find the following:

a. Find the exact value of k.

b. How much of the substance remains after 42 days.

c. When will there only be 20 grams remaining?

ASSIGNMENT: Page 338 #1-4, 9, 12, 13, 25

CALCULUS CHAPTER 6 ASSIGNMENT SHEET

SECTION 6-1 (Day 1) Indefinite Integrals

ASSIGNMENT: Page 312 #3 – 6, 9, 10, 13, 15, 17, 19, 22

SECTION 6-1 (Day 2) Solving Differential Equations

ASSIGNMENT: Page 313 #27, 29, 31 – 34, 36, 41, 42

SECTION 6-1 (Day 3) Slope Fields

ASSIGNMENT: Slope Fields Handout #1-16

SECTION 6-2 (Day 1) Substitution Method

ASSIGNMENT: Page 321 – 322 #2 – 4, 6, 8, 9, 13, 17, 24

SECTION 6-2 (Day 2) Substitution w/ Trig Functions

ASSIGNMENT: Page 321 – 322 #11, 14, 16, 18, 19, 21, 22, 34, 36, 37

SECTION 6-2 (Day 3) Separating Variables

ASSIGNMENT: Page 322 #42, 43, 44

SECTION 6-3 Integration by Parts

ASSIGNMENT: Page 328 # 2, 15, 16, 19

SECTION 6-4 Exponential Growth and Decay

ASSIGNMENT: Page 338 #1-4, 9, 12, 13, 25

CHAPTER SIX REVIEW SHEET

CHAPTER SIX REVIEW SHEET

CHAPTER SIX TEST