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11BASIC CALCULUSQuarter 4 – Module 1Antiderivative of a FunctionSENIOR HIGH SCHOOLBasic Calculus – Grade 11Alternative Delivery ModeQuarter 4 – Module 1: Antiderivative of a FunctionFirst Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in anywork of the Government...

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11

SENIOR HIGH SCHOOL

BASIC CALCULUS Quarter 4 – Module 1 Antiderivative of a Function

Basic Calculus – Grade 11 Alternative Delivery Mode Quarter 4 – Module 1: Antiderivative of a Function First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer: Mercyditha D. Enolpe Editors: Ronald G. Tolentino & Gil S. Dael Reviewer: Littie Beth S. Bernadez Layout Artist: Radhiya A. Ababon Management Team:

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11 Basic Calculus Quarter 4 – Module 1 Antiderivative of a Function

I LEARNING COMPETENCIES: ▪ Illustrate an antiderivative of a function (STEM_BC11I-IVa-1) ▪ Compute the general antiderivative of polynomial, radical, exponential, and trigonometric functions (STEM_BC11I-IVa-b-1) OBJECTIVES: K: Illustrate antiderivative of a function; S: Compute the general antiderivative of polynomial, radical, exponential and logarithmic functions; A: Develop perseverance in computing general antiderivative of polynomial, radical, exponential and logarithmic functions.

I PRE-ASSESSMENT Multiple Choice. Answer the following statements by writing the letter of the correct answer on your activity notebook/activity sheets. 1. In the expression dx, what does it tell you to integrate? A. d B. x C. y 2. What does k represent in the expression: ∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶? A. Kilometer B. Kelvin C. Derivative

2

D. u D. Constant

3. Using the theorem of antidifferentiation, how should the expression below be corrected to make it true? ∫(𝑓(𝑥) − 𝑔(𝑥))𝑑𝑥 = ∫ 𝑔(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑑𝑥

A. (-) sign and (+) are supposed to be ((÷) and (x), respectively. B. The (+) sign at the right side of the equation should be (-). C. There should have no integral sign, instead

𝑑𝑢

𝑑𝑥

should be used.

D. The (-) sign at the left side of the equation should be (+) and g(x) and f(x) at the right side of the equation are interchanged. 4. What word makes this statement wrong: “The integral of the product of a constant, a, and a function, f(x), is the quotient of the constant and the integral of the function.”? A. Integral

B. Quotient

C. Constant

D. Function

5. Which refers to the process of finding the antiderivative? A. Differentiation

B. Antidifferentiation C. Integration

D. Integration

6. What do you call the symbol ∫ ? A. Integrand

B. Radical sign

C. Derivative sign

D. Integral sign

7. What is the function f called in the expression 𝐹(𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 ? A. Integrand

B. Radical sign

8. Find the integral of ∫ 2 𝑑𝑥 . 2

𝑥2

C. Derivative sign

D. Integral sign

+𝐶

C. 2x +C

D. 𝑥 2 + 𝐶

+𝐶

C.

A. 𝑥 + 𝐶

B.

A. 4𝑥 4 + 𝐶

B.

A. sec x + C

B. cos x + C

2

9. What is its integral of In ∫ 𝑥 3 𝑑𝑥? 𝑥4 4

10. Find the integral of ∫ 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 𝑑𝑥 .

𝑥4 2

+𝐶

C. csc x + C

3

D. 3𝑥 + 𝐶

D. sin x + C

Lesson 1

Illustration of Antiderivative of a Function and Computing the General Antiderivatives of Functions ’s In

REVIEW To understand the ideas of antidifferentiation or getting the antiderivative of a function, let us look back our lesson in getting the derivatives of functions by answering the activity below. ACTIVITY 1 Matching Type. Match the functions in Column A with their corresponding derivatives in Column B. Write the letter of the correct answer in your activity sheet/notebook. Column A

Column B

1. 𝑓(𝑥) = 5𝑥3

A. 𝑓(𝑥) = 9

4. 𝑓(𝑥) = 9 x

D. 𝑓(𝑥) = 2𝑥

1

2. 𝑓(𝑥) = 𝑥2 +2

B. 𝑓(𝑥) = 3𝑥2 + 4𝑥 + 1

3. 𝑓(𝑥) = 𝑥3 + 2

C. 𝑓(𝑥) =15𝑥 2

1

5. 𝑓(𝑥) = 𝑥3 + 2𝑥 2 + 𝑥

E. 𝑓(𝑥) = 3𝑥2

In the previous discussion, we learned how to find the derivatives of different functions. Now, we will introduce the inverse of differentiation. We shall call this process as antidifferentiation. A natural question then arises: a. Given a function f, can we find a function F whose derivative is f? b. Is it possible to find a function 𝒚 = 𝑭(𝒙) for which f(x) is the derivative? This is the “anti” or inverse problem of finding the derivative. Thus the function F(x) is called antiderivative of f(x) if and only if F’(x)= f(x).

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’s New Situations: 1. College Connections Inc. is a scholarship-search service that helps high school students find college scholarships, in developing their marketing plan for the next year, the marketing manager of College Connections needs to determine the number of people that will be in the college-age bracket this year. They will use a life table function, a function l such that f(x) is the number of people in the population who reach the age of x years at any time during the year, to find this population. Life table functions involve integrals. To understand the concept of an integral we consider the following definitions, theorems and discussions below.

is It DISCUSSION Definition of Integral F(x) is an integral of f(x) with respect to x if and only if F(x) is an antiderivative of f(x). That is, F(x) = ∫ 𝑓(𝑥)𝑑𝑥 if and only if F’(x)=f(x) TERMINOLOGIES AND NOTATIONS: • Antidifferentiation is the process of finding the antiderivative. • The symbol ∫ , also called the integral sign, denotes the operation of antidifferentiation. • The function f is called integrand. • If F is an antiderivative of f, we write ∫ 𝒇(𝒙)𝒅𝒙 = 𝑭(𝒙) + 𝑪. •

•

The symbols ∫ and dx go hand-in-hand and dx helps to identify the variable of integration. The expression F(x) + C is called the general antiderivative of f. Meanwhile, each antiderivative of f is called a particular antiderivative of f.

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BASIC THEOREMS ON ANTIDIFFERENTIATION (a) If k is a constant, ∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶

(b) If n is any real number and n ≠ −1. 𝑡ℎ𝑒𝑛 ∫ 𝑥 𝑛 𝑑𝑥 =

𝑥𝑛+1

𝑛+1

+ C where x is a differentiable

function. (c) The integral of a sum of functions is the sum of the integrals of the functions. ∫(𝑓(𝑥 ) + 𝑔(𝑥))𝑑𝑥 = ∫ 𝑓(𝑥 )𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥

(d) The integral of the product of a constant, a, and a function, f(x), is the product of the constant and the integral of the function. ∫ 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫ 𝑓(𝑥)𝑑𝑥.

Note the using (c), we can have ∫[𝑓(𝑥 ) − 𝑔(𝑥 )]𝑑𝑥 = ∫ 𝑓(𝑥 )𝑑𝑥 − ∫ 𝑔(𝑥 )𝑑𝑥 .

Examples:

1. If k is a constant, ∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶 1.a. Find the integral of ∫ 1𝑑𝑥 = ∫ 𝑑𝑥 Solution: Since 𝐹 ′ (𝑋) = 1 when 𝐹(𝑥) = 𝑥 + 𝐶, then ∫ 1𝑑𝑥 = 𝑥 + 𝐶.

1.b. Find the integral of ∫ 5 𝑑𝑥. Solution: Since 𝐹 ′ (𝑋) = 5 when 𝐹(𝑥) = 5𝑥 + 𝐶, then ∫ 5 𝑑𝑥 = 5𝑥 + 𝐶. 1.c. Find the integral of ∫ 3 𝑑𝑥. Using (a) of the theorem, we ∫ 3 𝑑𝑥 = 3𝑥 + C.

2. If n ≠ −1. ∫ 𝑥 𝑛 𝑑𝑥 =

𝑥𝑛+1

𝑛+1

+ C where x is a differentiable function.

2.a. Find the integral of ∫ 𝑥 2 𝑑𝑥.

Solution: Since 𝐹 ′ (𝑋) = 𝑥 2 when 𝐹(𝑥) = then ∫ 𝑥 2 𝑑𝑥 =

𝑥3 3

+ 𝐶.

2.b. Find the integral of ∫ 𝑥 5 𝑑𝑥 .

Solution: Since 𝐹 ′ (𝑋) = 𝑥 5 when 𝐹(𝑥) = then ∫ 𝑥 5 𝑑𝑥 =

𝑥6 6

+ 𝐶.

2.c. Determine the antiderivative ∫ 𝑥 6 𝑑𝑥.

𝑥2+1

+ 𝐶, which is equal to (𝑥) =

𝑥3

+ 𝐶,

𝑥5+1

+ 𝐶, which is equal to (𝑥) =

𝑥6

+ 𝐶,

2+1

5+1

Using (b) of the theorem, we have ∫ 𝑥 6 𝑑𝑥 =

𝑥6+1

6+1

+𝐶 =

𝑥7 7

3

6

+ 𝐶.

3. If f and g are functions. The integral of a sum of functions is the sum of the integrals of the functions. ∫(𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥 6

3.a. Find the antiderivative of ∫(4𝑥 +7)dx. Solution: Use the formula that says the integral of a sum of functions is the sum of the integrals. ∫(4𝑥 + 7)𝑑𝑥 = ∫ 4𝑥𝑑𝑥 + ∫ 7𝑑𝑥

= 4 ∫ 𝑥𝑑𝑥 + ∫ 7𝑑𝑥

[Using (d)]

= 4 ( 2 + 𝐶1 ) + 7𝑥 + 𝐶2

[Use the (b) and (a) formula, respectively)]

= 2𝑥 2 + 7𝑥 + 4𝐶1 + 𝐶2

[let 𝐶 = 4𝐶1 + 𝐶2 ]

𝑥2

= 2𝑥 2 + 4𝐶1 + 7𝑥 + 𝐶2 = 2𝑥 2 + 7𝑥 + 𝐶.

Thus,

∫(4𝑥 + 7)𝑑𝑥 = 2𝑥 2 + 7𝑥 + 𝐶 A. Computing Antiderivatives of Polynomial Functions Illustration 1: a. An antiderivative of 𝑓(𝑥) = 12𝑥 2 + 2𝑥 is 𝐹(𝑥) = 4𝑥 3 + 𝑥 2 . As we can see, the derivative of F is given by 𝐹 ′ (𝑥) = 12𝑥 2 + 2𝑥 = 𝑓 (𝑥). b. Find the integral of ∫(3𝑥 2 + 6𝑥 + 1)𝑑𝑥 Solution: ∫(3𝑥 2 + 6𝑥 + 1)𝑑𝑥 = ∫ 3𝑥 2 𝑑𝑥 + ∫ 6𝑥𝑑𝑥 + ∫ 1𝑑𝑥

= 3 ∫ 𝑥 2 𝑑𝑥 + 6 ∫ 𝑥𝑑𝑥 + 1 ∫ 𝑑𝑥. = 3( 3 + 𝐶1 ) + 6( 𝑥3

𝑥2

2

+ 𝐶2 ) + (𝑥 + 𝐶3 )

= 𝑥 3 + 3𝐶1 + 3𝑥 2 + 6𝐶2 + 𝑥 + 𝐶3 = 𝑥 3 + 3𝑥 2 + 𝑥 + 3𝐶1 + 6𝐶2 + 𝐶3

Thus,

= 𝑥 3 + 3𝑥 2 + 𝑥 + 𝐶

∫ 3𝑥 2 + 6𝑥 + 1)𝑑𝑥 = 𝑥 3 + 3𝑥 2 + 𝑥 + 𝐶. B. Computing Antiderivatives of Radical Functions 1. Find the antiderivative of f(x)= √x3 . Solution:

7

by (c) by (d) by (b) and (a)

(let C = 3𝐶1 + 6𝐶2 + 𝐶3 ))

Note that the expression √𝑥 3 can be written as 𝑥 2 using rational exponents using (b), we get 3

f(x)= √𝑥 3 .=∫ 𝑥 2 𝑑𝑥 3

=

=

2. 𝑓(𝑥) = √𝑥 2. Solution: 3

3 2

𝑥 2+2 3 2 + 2 2 5

𝑥2 5 2

+𝐶

+ 𝐶 = 5 𝑥 2 + C or 5 √𝑥 5 + 𝐶. 2

5

2

2

Rewriting √𝑥 2 as 𝑥3 and using (b), we get, 3

∫ √𝑥 2 𝑑𝑥 =∫ 𝑥 3 𝑑𝑥 2

3

=

=

2 3

+ 𝑥3 3 2 3 + 3 3 5

𝑥3 5 3

+𝐶

+ 𝐶 = 𝑥 3 +C or 5 3

5

2 3 5 √𝑥 +C. 5

C. Computing Antiderivatives of Exponential Functions Exponential functions can be integrated using the following formulas. ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶

∫ 𝑎𝑥 𝑑𝑥 =

𝑎𝑥 +𝐶 𝑙𝑛 𝑎

D. Computing Antiderivatives of Trigonometric Functions The Antiderivatives of Basic Trigonometric Functions Recall from the definition of an antiderivative that, if 𝑑 𝑓(𝑥) 𝑑𝑥

= 𝑔 (𝑥) , then g(x) dx = f(x) + C.

8

That is, every time we have a differentiation formula, we get an integration formula for nothing. Here is a list of some of them. DERIVATIVE Rule 𝑑 sin 𝑥 = cos 𝑥 𝑑𝑥

ANTIDERIVATIVE Rule ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

𝑑 cos 𝑥 = −𝑠𝑖𝑛 𝑥 𝑑𝑥

∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶

𝑑 cotan 𝑥 = − 𝑐𝑜𝑠𝑒𝑐 2 𝑥 𝑑𝑥

∫ 𝑐𝑠𝑐 2 𝑥 𝑑𝑥 = −cot 𝑥 + 𝐶

𝑑 tan 𝑥 = 𝑠𝑒𝑐 2 𝑥 𝑑𝑥

𝑑 sec 𝑥 = sec 𝑥 tan 𝑥 𝑑𝑥

𝑑 csc 𝑥 = −𝑐𝑠𝑐𝑥 cot 𝑥 𝑑𝑥

∫ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶

∫(sec 𝑥 tan 𝑥) 𝑑𝑥 = sec 𝑥 + 𝐶

∫(csc 𝑥 cot 𝑥) 𝑑𝑥 = − csc 𝑥 + 𝐶

Notice that, quite by chance, we have come up with formulas for the antiderivatives of sin x and cos x. What about the other four? Here they are!

∫ tan(𝑥 ) 𝑑𝑥 = −𝑙𝑛 |cos(𝑥 )| + 𝐶 = 𝑙𝑛 |sec (𝑥)|+C ∫ cot(𝑥 ) 𝑑𝑥 = 𝑙𝑛|sin(𝑥)| + 𝐶 = −𝑙𝑛|csc (𝑥)|+C ∫ sec(𝑥) 𝑑𝑥 = 𝑙𝑛|sec(𝑥) + tan (𝑥)| + 𝐶

∫ csc(𝑥) 𝑑𝑥 = −𝑙𝑛|csc(𝑥) + cot (𝑥)| + 𝐶 In computing for the antiderivatives of trigonometric functions, let us assume knowledge of the Basic Theorems on Antidifferentiation. Most of the problems are average. A few are challenging. Many use the method of u-substitution. Make careful and precise use of the differential notation dx and du and be careful also when arithmetically and algebraically simplifying expressions.

9

Examples:

1. Compute ∫(3 sin 𝑥 − 4𝑠𝑒𝑐 2 𝑥) 𝑑𝑥

Solution:

∫(3 sin 𝑥 − 4𝑠𝑒𝑐 2 𝑥) 𝑑𝑥

= ∫ 3 sin 𝑥 𝑑𝑥 − ∫ 4𝑠𝑒𝑐 2𝑥)𝑑𝑥 = 3 ∫ sin 𝑥 𝑑𝑥 − 4 ∫ 𝑠𝑒𝑐 2𝑥 𝑑𝑥

= 3 (- cos x) - 4 tan x + C

2. Evaluate ∫(3 𝑐𝑜𝑠 𝑥 + 𝑥 2 ) 𝑑𝑥 .

using (c) using (a) (from the table)

Solution:

∫(3 𝑐𝑜𝑠 𝑥 + 𝑥 2 ) 𝑑𝑥 = ∫ 3 𝑐𝑜𝑠 𝑑𝑥 + ∫ 𝑥 2 𝑑𝑥

= 3 ∫ 𝑐𝑜𝑠 𝑑𝑥 + ∫ 𝑥 2 𝑑𝑥

𝑥 2+1 = 3 𝑠𝑖𝑛 𝑥 + +𝐶 2+1 𝑥3 = 3 𝑠𝑖𝑛 𝑥 + + 𝐶 3

’s More ENRICHMENT ACTIVITY

1. ∫ 𝑥6 𝑑𝑥

A. Determine the following antiderivatives. 1

2. ∫ 4√𝑢du

3. ∫(12𝑥 2 + 2𝑥)𝑑𝑥

4. ∫

𝑥 2 +1 𝑥2

dx

10

using (c)

using (a)

using the formula in the box and (b)

Solution: Using the Theorems on antidifferentiation to determine the antiderivatives. 1. Using (b) of the theorem, we have 1 𝑥 −6+1 1 𝑥 −5 ∫ 6 𝑑𝑥 = ∫ 𝑥 −6 𝑑𝑥 = +𝐶 =− 5+𝐶 +𝐶 = 5𝑥 𝑥 −5 −6 + 1

2. Using (b) and (c) of the theorem, we have ∫ 4√𝑢 𝑑𝑢 =

1 4 ∫ 𝑢 2 𝑑𝑢

1

𝑢 2+1 )+𝐶 = 4( 1 + 1 2 3

𝑢2 = 4( )+𝐶 3 2

2 3 = 4 ( ) 𝑢2 + 𝐶 3 83 = 𝑢2 + 𝐶 3 3

2 8𝑢 = +𝐶 3

3. Using (b), (c) and (d) of the theorem, we have

∫(12𝑥 2 + 2𝑥)𝑑𝑥 = 12 ∫ 𝑥 2 𝑑𝑥 + 2 ∫ 𝑥𝑑𝑥 = 12 (

𝑥2 𝑥3 ) + 2 ( ) + 𝐶 = 4𝑥 3 + 𝑥 2 + 𝐶 2 3

4. Using (a), (b), and (d), we have 1 1 𝑥 −1 𝑥2 + 1 + 𝐶 = 𝑥 − + 𝐶. ∫ 𝑑𝑥 = ∫ (1 + ) 𝑑𝑥 = 𝑥 + 2 2 −1 𝑥 𝑥 𝑥 Independent Activity Look back and reflect. 1. How do you compute antiderivatives of functions? 2. How do you know that f(x) can be integrated? 3. What is the difference between differentiation and integration?

11

I Have Learned Generalization Directions: Reflect the learning that you gained after taking up the two lessons in this module by completing the given statements below. Do this on your activity notebook. Do not write anything on this module. What were your thoughts or ideas about the topic before taking up the lesson? I thought that _____________________________________________________________ __________________________________________________________________________. What new or additional ideas have you had after taking up this lesson? I learned that (write as many as you can) ___________________________________________________________________________ _______________________________________________________________________. How are you going to apply your learning from this lesson? I will apply ___________________________________________________________________________ _______________________________________________________________________.

12

I Can Do Application (Performance Task) Direction: Give at least 1 (one) example with complete solution of each of the Basic Theorems in Antidifferentiation and an example also of each of the Antiderivatives of Trigonometric Functions. Each of the example is worth 2 points. The maximum possible points earned on this task is 35. Examples should not be copied from this module. A rubric is provided as basis in grading the output. Write your answer in the drawing book that you used during the Third Quarter on Limit Laws. RUBRIC on EXAMPLES OF ANTIDIFFERENTIATION CATEGORY 5 4 3 2 The The project’s The project’s The project’s Appearance appearance is appearance is appearance of the project project’s appearance quite clean and somewhat is quite poor. poor. Missing Does not is clean and with only few unnecessary some of the include the free from required proper unnecessary marks. parts. requirements marks. in the task. Application of appropriate steps in solving the problem

Uses completely appropriate steps in solving

Applies some appropriate steps in solving

Applies inappropriate steps in solving

Correctness of the answers/ examples

The answers/ examples are correct.

The answers/example has a slight mistake.

Copying Has shown error, answer but computational incorrect error , Just presented partial answer only

Applies No inappropriate answer steps in at all solving which produces wrong answer

Note: Each example presented will credit 2 points even if it’s incorrect. Basic Theorems in Antidifferentiation (4 x 2 ) = 8. Antiderivatives of Trigonometric Functions (6 x 2) = 12. Points from the rubric (highest possible is 15). Total possible score is 35. 13

1 Has not shown any of the parts as required in the task.

No answer at all

Multiple Choice. Read and understand each statement. Write the letter of the correct answer on your activity notebook. 1. Evaluate∫(3𝑥 + 2)𝑑𝑥 A. 3𝑥 2 + 2𝑥 + 𝐶 B. 2𝑥 2 + 2𝑥 + 𝐶

C.

3 2 𝑥 2

+ 2𝑥 + 𝐶

D.

1 2 𝑥 2

+ 2𝑥 + 𝐶

2. Evaluate∫ 𝑑𝑥. A. 𝑥 + 𝐶 B. 𝑥 C. −𝑥 + 𝐶 D. −𝑥 3. Addition: Subtraction as _________: Antidifferentiation A. Integration B. Optimization C. Differentiation D. Abstraction 4. Which of the symbols below denote the operation of antidifferentiation? 𝑑 C. ∫ 𝑓(𝑥 )𝑑𝑥 = 𝐹(𝑥 ) A. (𝑓(𝑥)) = 𝐹(𝑥) + 𝐶 𝑑𝑥

D. ∫ 𝑓(𝑥 )𝑑𝑥 = 𝐹(𝑥 ) + 𝐶

B. 𝑓′(𝑥) = 𝐹 (𝑥) + 𝐶

5. In antidifferentiation, the formula ∫ 𝑥 𝑛 𝑑𝑥 =

𝑥𝑛

(𝑛+1)

+ 𝐶is valid if?

A. 𝑛 ∈ ℝ B. 𝑛 ≠ 1, 𝑛 ∈ ℝ C. 𝑛 ∈ {0,1,2} D. 𝑛 ∉ ℝ 3 6. In item number 5, the integral of ∫ 𝑥 𝑑𝑥 is obtained by applying what Basic Theorem of Antidifferentiation? C. Theorem a C. Theorem b D. Theorems a, b, c, d D. Theorems b and c 7. Find ∫ 6𝑥 𝑑𝑥? 𝑥2 A. 𝑥 2 + 𝐶 B. 2𝑥 2 + 𝐶 D. 3𝑥 2 + 𝐶 C. +𝐶 3

8. To find an antiderivative of a constant times a function, set aside the constant and find the antiderivative of the functions separately, then ______ the result. A. multiplu B. subtract C. add D. divide 9. If 𝐹(𝑥) = 2𝑥 2 + 5𝑥 + 2, then 𝐹 ′ (𝑥) = 4𝑥 + 5.Thus, if 𝑓(𝑥) = 4𝑥 + 5, then f is the ___ of F and thus F is a/an ________ of f. A. derivative, antiderivative C. antiderivative, inverse B. derivative, inverse D. antiderivative, derivative 10. Which statement below is NOT a reason of putting “+ C” at the end of an integration? A. It allows us to express the general form of antiderivatives. B. C is a constant of any value. C. Added at the end of an antiderivative which indicates indefinite integral of f(x). D. C is very important in integration because an answer of an antiderivative without the constant C means zero.

14

15 WHAT I CAN DO:

2.A

7. D

3.C

8. A

4.D

9. A
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