Basic-Calculus Q4-Week-1 Module-9 PDF

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Basic CalculusQuarter 4 – Week 1 Module 9Antiderivative of a FunctionSHS####### Basic Calculus####### Grade 11 Quarter 4 – Week 1 Module 8: Antiderivative of a Function####### First Edition, 2021####### Copyright © 2021####### La Union Schools Division####### Region I####### All rights reserved. No ...

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Basic Calculus Quarter 4 – Week 1 Module 9 Antiderivative of a Function

Basic Calculus Grade 11 Quarter 4 – Week 1 Module 8: Antiderivative of a Function First Edition, 2021

Copyright © 2021 La Union Schools Division Region I

All rights reserved. No part of this module may be reproduced in any form without written permission from the copyright owners.

Development Team of the Module

Author: Jiezyl Jamaica M. Aquino, T-II Editor: SDO La Union, Learning Resource Quality Assurance Team Illustrator: Ernesto F. Ramos Jr., P II

Management Team:

ATTY. Donato D. Balderas, Jr. Schools Division Superintendent Vivian Luz S. Pagatpatan, PHD Assistant Schools Division Superintendent German E. Flora, PHD, CID Chief Virgilio C. Boado, PHD, EPS in Charge of LRMS Erlinda M. Dela Peña, EDD, EPS in Charge of Mathematics Michael Jason D. Morales, PDO II Claire P. Toluyen, Librarian II

Target This learning material will provide you with information and activities that will deepen your understanding about Antiderivative of a Function. After going through this module, you are expected to:

▪ Illustrate an antiderivative of a function STEM_BC11l-Iva-1 ▪ Compute the general antiderivative of polynomial, radical, exponential, and trigonometric functions STEM_BC11l-IVab-1

Before going on, check how much you know about this topic. Answer the pre-test in a separate sheet of paper.

Pre-Test Directions: Read carefully each item. Use a separate sheet for your answers. Write only the letter of the best answer for each test item. 1. What do you call this symbol ∫? A. Arbitrary constant B. Euler number C. Integral sign D. Integrand 2. What do you call the 𝑓 in the function∫ 𝑓(𝑥)𝑑𝑥? A. Arbitrary constant B. Integral sign C. Integrand

D. Variable of integration

3. What do you call the 𝑑𝑥 in the function ∫ 𝑓(𝑥)𝑑𝑥? A. Arbitrary constant B. Integral sign C. Integrand D. Variable of integration 4. What do you call the 𝐶 in the antiderivative 𝑥 + 𝐶 ? A. Arbitrary constant B. Integral sign C. Integrand D. Variable of integration 5. What do you call the process of finding the general antiderivative of a given function? A. Chain rule B. Differentiation C. Integration D. Optimization

6. The following are functions that can be best integrated using Constant Rule, EXCEPT? A. ∫ 𝑑𝑥 B. ∫ 5𝑑𝑥 C. ∫ 𝑥𝑑𝑥 D. ∫ 10𝑑𝑥 7. Which of the following functions is BEST integrated using the Constant Multiple Rule? A. ∫ 9𝑑𝑥 B. ∫ 3𝑥8 𝑑𝑥 C. ∫ 𝑥 −3 𝑑𝑥 D. ∫(3𝑥 + 9)𝑑𝑥 8. The following are functions that are collectively called transcendental functions, EXCEPT? A. Algebraic functions B. Exponential functions C. Logarithmic functions D. Trigonometric functions 9. Which of the following is the ∫ 𝑒 𝑥 𝑑𝑥? A. 𝑥 + 𝐶 B. 𝑒 𝑥 + 𝐶 C. 𝑙𝑛|𝑥| + 𝐶 10. Which of the following is the ∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥? A. sin 𝑥 + 𝐶 B.cos 𝑥 + 𝐶 C. − cot 𝑥 + 𝐶 For numbers 11 – 15. Evaluate the following integrals. 11. ∫ 2𝑥3 𝑑𝑥 A.

12. ∫ 20𝑥 𝑑𝑥 13. ∫

10 𝑑𝑥 𝑥

𝑥4 2

+𝐶

B.

𝑥4 4

1

+𝐶

A. 20𝑥 + 𝐶

B.

A. 10 ln|𝑥| + 𝐶

B. 10𝑥 ln|𝑥| + 𝐶

+𝐶

𝑑𝑥 A. 4 cot 𝑥 + 𝐶 B. −4 cot 𝑥 + 𝐶 15. ∫ 8 tan 𝑥 𝑑𝑥 A. −8 ln|cos 𝑥| + 𝐶 C. −8 ln|csc 𝑥| + 𝐶

14. ∫

4𝑐𝑠𝑐2 𝑥

20𝑥

C. 2𝑥4 + 𝐶 20𝑥

C. ln 20 + 𝐶

C.

1 ln|𝑥| 10

D. sin 𝑥 + 𝐶 D. − csc 𝑥 + 𝐶 D. 4𝑥4 + 𝐶 D.

+𝐶

C. 4 sec 𝑥 + 𝐶 B. −8 ln|sin 𝑥| + 𝐶 D. −8 ln|sec 𝑥| + 𝐶

D.

ln 20 20𝑥

+𝐶

1 𝑥 ln|𝑥| 10

+𝐶

D. −4 sec 𝑥 + 𝐶

Antiderivative of a Function

1

Jumpstart

For you to understand the lesson well, do the following activities. Have fun and good luck! Activity 1: Match Me! Directions: Match each of the indefinite integral to its result, where 𝐶 is a constant. Choose the correct answer in the given box below. Then, decipher the hidden word. Use a separate sheet of paper for your answers. A. 2𝑥 + 𝐶 B. 2𝑥2 + 5𝑥 + 𝐶

E.3𝑥3 − 7𝑥 + 𝐶 F. 𝑥 3 + 4𝑥 2 + 𝐶

C. 2𝑥3 + 5𝑥2 + 𝑥 + 𝐶

I. 𝑥 2 + 𝐶 J. 𝑥 + 𝐶

G. 3𝑥2 + 7𝑥 2 + 𝐶

D. 𝑥 3 − 4𝑥2 + 𝐶

K.

H. 3𝑥2 + 7𝑥 + 𝐶

L.

3 1 3 𝑥 + 𝑥 2 − 9𝑥 + 𝐶 2 2 1 5 3 2 𝑥 + 𝑥 − 9𝑥 + 𝐶 5 2

INTEGRAL ∫ 2𝑥 𝑑𝑥

_____________1.

∫(4𝑥 + 5)𝑑𝑥

_____________2.

∫(3𝑥2 − 8)𝑑𝑥

_____________3.

∫(9𝑥2 − 7)𝑑𝑥

_____________4.

∫ 𝑥4 + 3𝑥 − 9𝑑𝑥

_____________5.

N 1

3

4

5

1

2

5

4

Discover 𝑓

A function F is called an antiderivative of a function for every value o in .

on an interva

i 𝐹

=

Antiderivatives are always denoted with a capital letter using the explicit rule for functions. This means that the antiderivative of 𝑓(𝑥) is written as 𝐹(𝑥), and the antiderivative of 𝑔(𝑥) is written as 𝐺(𝑥). Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals. Indefinite Integrals (also called antiderivatives) do not have limits/bounds of integration, while definite integrals do have bounds. 9

∫ 𝑥2

∫0 𝑥 2

Indefinite integral No bounds

Definite integral Bounded from 0 to 9

When you find an indefinite integral, you always add a “+ C” (called the constant of integration) to the solution. The “+ C” indicates that the solution actually has infinite possibilities. That’s because you can have many solutions, all of which are the set of all vertical transformations of the antiderivative. For example, the antiderivative of 2x is x 2 + C, where C is a constant. The derivative of a constant is zero, so C can be any constant, positive or negative. Four antiderivatives of 2x are x2 + 1, x2 -1, x2 + 2 or x2 – 2. Antiderivative or integrals

2x

𝑥2 + 2 Derivatives

𝑥2 + 1

𝑥2 – 1

𝑥2 – 2

Because a single continuous function has infinitely many antiderivatives, we do not refer to "the antiderivative", but rather, a "family" of antiderivatives, each of which differs by a constant. So, if F is an antiderivative of f, then G = F + c is also an antiderivative of f, and F and G are in the same family of antiderivatives.

The notation used to refer to antiderivatives is the indefinite integral. f (x)dx means the antiderivative of f with respect to x. If F is an antiderivative of f, we can write

f (x)dx = F + c. In this context, c is called the constant of integration.

To find antiderivatives of basic functions, the following rules can be used: Rule

Function =Integral ∫ 𝒙 𝒅𝒙 =

Power Rule (n≠ 1)

∫𝒌

Multiplication by Constant Rule

𝒙 𝟏 +𝑪 𝒏+𝟏

𝒅𝒙 = 𝒌 ∫

𝒅

Sum Rule

∫

+

𝒅𝒙 = ∫

𝒅𝒙 + ∫

𝒅

Difference Rule

∫

−

𝒅𝒙 = ∫

𝒅𝒙 − ∫

𝒅

Antiderivative of a polynomial functions Example 1.Find the antiderivative of 𝒇(𝒙) = 𝒙𝟐 Solution:

Step 1: Use Power Rule. Plugging our term in the formula ∫ 𝒙𝒏 = = 2 we have 𝑥 2+1 ∫ 𝑥2 𝑑𝑥 = +𝐶 2+1 3 𝑥 +C = ∫ 𝒙 𝒅𝒙

𝟏 𝟑 𝒙 𝟑

𝒙𝒏+𝟏 𝒏+𝟏

+ 𝑪, where n

+C

𝟏

The antiderivative of 𝒇(𝒙) = 𝒙𝟐 is F(x) = 𝟑 𝒙𝟑 + c Example 2: Evaluate ∫ 4𝑥3 𝑑𝑥 Solution: Step1. Use Multiplication by Constant Rule. We can move the 4 outside the integral sign, we have ∫ 4𝑥 3 𝑑𝑥 = 4 ∫ 𝑥 3 𝑑𝑥 Step 2. Plugging our term in the Power Rule where n = 3

𝑥 3+1

∫ 4𝑥3 𝑑𝑥 = 4 [

3+1

= (4)

Step 3: Simplify

∫ 4𝑥3 𝑑𝑥 = 𝒙𝟒 + 𝑪

]+𝐶

𝑥4 4

+𝐶

Example 3:Evaluate ∫(𝟑𝒙 + 𝟓) 𝒅𝒙

Solution:

Step 1. Because this equation consists of terms added together, you can integrate them separately and add the results, giving us:

∫(3𝑥 + 5)𝑑𝑥 = ∫ 3𝑥𝑑𝑥 + ∫ 5𝑑𝑥 Step 2: Use Multiplication by Constant Rule = 3 ∫ 𝑥 𝑑𝑥 + 5 ∫ 𝑑𝑥

Step 3: We know the fact that 𝑥 0 = 1, we have

= 3 ∫ 𝑥 𝑑𝑥 + 5 ∫ 𝑥 0 𝑑𝑥

Step 4: Plugging our term in the Power Rule where n = 1 and n = 0 simultaneously, we have, 𝑥 0+1

𝑥 1+1

= 3 [ 1+1 ] + 5[ 0+1 ] + 𝐶 =

Step 5: Simplify

3𝑥2 5𝑥1 +𝐶 + 1 2

𝟑 𝟐 𝒙 + 𝟓𝒙 + 𝑪 𝟐

∫(𝟑𝒙 + 𝟓)𝒅𝒙 =

Example 4: Find the antiderivative of f(x) = 𝑥 −3

Solution: Step 1. Plugging our term in the Power Rule, where n = -3, we have ∫𝒙

=

𝒙−𝟑+𝟏 −𝟑+𝟏

+C

=

𝒙−𝟐

Step 2. We know the fact that 𝑥 −2 =

1 , 𝑥2

+C

−𝟐

=-

we have

𝟏 𝟐𝒙𝟐

+C

Antiderivative of a radical functions Example 5. Evaluate ∫

3 𝑥2 +4 √ 𝑥

𝑥

𝑑𝑥

Solution: Step 1. Rewrite the integrand and simplify, we have 3 𝑥2 +4 √ 𝑥

𝑥

=

𝑥2 𝑥

+

43√𝑥 𝑥

=

𝑥2

+

𝑥

4𝑥1/3 𝑥

1

= 𝑥 2−1 + 4𝑥 3−1 = x + 4𝑥 −2/3 Step 2. Integrate each term of these terms separately, we have, ∫(x + 4𝑥 −2/3) dx = ∫ 𝑥𝑑𝑥 + ∫ 4𝑥 −2/3dx Step 3. Use Multiplication by Constant Rule = ∫ 𝑥𝑑𝑥 + 4 ∫ 𝑥 −2/3dx 2

Step 4.Use Power Rule where n = 1 and n = - 3 simultaneously, we have =

2

𝑥1+1

+ 4[

− +1 𝑥 3

𝑥2

+ 4[

𝑥1/3

1+1

=

2

−2 +1 3 1 3

]

]

Step 5. Simplify ∫

𝑥2 +4 3√𝑥 𝑥

𝑑𝑥 =

1

2

1

𝑥 2 + 12𝑥3 + C

Integral (antiderivative) of exponential function Common functions Exponential

Integral (Antiderivative) ∫ 𝑒 𝑑𝑥 = 𝑒 𝑥 + ∫ 𝑎 𝑑𝑥 =

𝑎𝑥 + 𝑙 𝑎

1 𝑑𝑥 = 𝑙 + 𝑜 𝑥≠ 𝑥 ∫ln(x) dx = x ln(x) − x + C ∫

Example 6: Evaluate ∫ 𝟕𝒙𝒅𝒙 Solution: Use the formula ∫ 𝑎 𝑑𝑥 =

𝑎𝑥 𝑙

+

where a = 7, we have ∫ 7𝑥 𝑑𝑥 =

𝟕𝒙 +𝑪 𝐥𝐧 𝟕

Example 7: Evaluate ∫ 𝟐𝒙+𝟑𝒅𝒙 Solution: Step 1. Transform the given expression using the laws of exponents. 2𝑥+3 = 2𝑥 . 23 = 𝟖 . 𝟐𝒙 Thus, ∫ 2𝑥+3 𝑑𝑥 = ∫ 8. 2𝑥 𝑑𝑥 Step 2. We can move the 8 outside the integral sign, we have = 8 ∫ 2𝑥 𝑑𝑥 Step 3. Use the formula ∫ 𝑎 𝑑𝑥 = 𝑙

𝑎𝑥

+

where a = 2, we have 2𝑥

= 8. ln 2 + 𝐶

8. 2𝑥 +𝐶 ln 2 Step 4. We know the fact that 2𝑥+3 = 8. 2𝑥 , hence 𝟐𝒙+𝟑 +𝑪 ∫ 𝟐𝒙+𝟑𝒅𝒙 = 𝐥𝐧 𝟐 =

𝟗

Example 8: Evaluate ∫ 𝒙 𝒅𝒙

Solution: Step 1. Rewrite the integral as

1 9 𝑑𝑥 = ∫ (9. ) 𝑑𝑥 𝑥 𝑥 Step 2. We can move the 9 outside the integral sign, we have ∫

1 = 9 ∫ 𝑥 𝑑𝑥 + 𝑓𝑜 𝑥 ≠ 0

Step 3. Use the formula ∫ 𝑑𝑥 = 𝑙 𝟏

𝟗

Example 9: Evaluate ∫ (𝒆𝒙 − 𝟕𝒙) 𝒅𝒙

∫ 𝒙 𝒅𝒙 = 𝟗 𝒍𝒏|𝒙| + 𝑪

Solution: Step 1. Using the difference rule of integration, we have 1 1 ∫ (𝑒 𝑥 − ) 𝑑𝑥 = ∫ 𝑒 𝑥 𝑑𝑥 − ∫ 𝑑𝑥 . 7𝑥 7𝑥 1

Step 2. We can move the 7 outside the integral sign, we have 1

1

= ∫ 𝑒 𝑥 𝑑𝑥 − ∫ 𝑥 𝑑𝑥 7

Step 3. Use the formula ∫ 𝑒 𝑑𝑥 = 𝑒 𝑥 + 𝐶 and ∫ 𝑑𝑥 = 𝑙

simultaneously, we have

∫ (𝒆𝒙 −

+

𝑓𝑜 𝑥 ≠

𝟏 𝟏 ) 𝒅𝒙 = 𝒆𝒙 − 𝒍𝒏|𝒙| + 𝑪 𝟕 𝟕𝒙

Integral (antiderivative) of trigonometric function

Trigonometric function

Integral (Antiderivative) ∫ 𝑠𝑖

∫ 𝑠𝑒

∫ 𝒕𝒂

𝑑𝑥 = −𝑐𝑜 𝑥 +

∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

𝑑𝑥 = 𝑡𝑎 𝑥 +

𝒅𝒙 = −𝒍 𝒄𝒐 ∫ 𝑠𝑒

𝑡𝑎

∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡 𝑥 + 𝐶 +

∫ 𝒄𝒐

𝒅𝒙 = 𝒍 𝒔𝒊

+

𝑑𝑥 = 𝑠𝑒 𝑥 + 𝐶

∫ csc 𝑥 cot 𝑥 𝑑𝑥 = −𝑐𝑠𝑐 𝑥 + 𝐶 ∫ 𝒔𝒆

∫ 𝒄𝒔

𝒅𝒙 = 𝒍 𝒔𝒆 𝒙 + 𝒕𝒂

𝒅𝒙 = −𝒍 𝒄𝒔 𝒙 − 𝒄𝒐

Example 10: Evaluate ∫(𝒔𝒊𝒏 𝒙 + 𝒄𝒐𝒔 𝒙)𝒅𝒙 Solution: Step 1: Sum rule for integrals, we have ∫(sin 𝑥 + cos 𝑥)𝑑𝑥 = ∫ sin 𝑥 𝑑𝑥 + ∫ cos 𝑥 𝑑𝑥. Step 2: Use the formula ∫ 𝑠𝑖 𝑑𝑥 = −𝑐𝑜 𝑥 + and ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶 simultaneously, we have ∫ sin 𝑥 𝑑𝑥 + ∫ cos 𝑥 𝑑𝑥. = − cos 𝑥 + sin 𝑥 + 𝐶 ∫(𝒔𝒊𝒏 𝒙 + 𝒄𝒐𝒔 𝒙)𝒅𝒙 = 𝐬𝐢𝐧 𝒙 − 𝐜𝐨𝐬 𝒙 + 𝑪

+

+

Example 11. Evaluate ∫(𝟒𝒄𝒔𝒄𝟐 𝒙 − 𝟑𝒔𝒆𝒄𝟐 𝒙) 𝒅𝒙 Solution: Step 1. Difference Rule for integral, we have ∫(4𝑐𝑠𝑐2 𝑥 − 3𝑠𝑒𝑐2 𝑥)𝑑𝑥 = ∫ 4𝑐𝑠𝑐2 𝑥𝑑𝑥 - ∫ 3𝑠𝑒𝑐2 𝑥𝑑𝑥 Step 2. We can move 4 and 3 outside the integral sign 4 ∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥 − 3 ∫ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥

Step 3. Use the formula ∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡 𝑥 + 𝐶 an ∫ 𝑠𝑒 = 4(− cot 𝑥) − 3(tan 𝑥 + 𝐶)

𝑑𝑥 = 𝑡𝑎 𝑥 + 𝐶

∫(𝟒𝒄𝒔𝒄𝟐 𝒙 − 𝟑𝒔𝒆𝒄𝟐 𝒙)𝒅𝒙 = −𝟒 𝐜𝐨𝐭 𝒙 − 𝟑 𝐭𝐚𝐧 𝒙 + 𝑪

Example 12: Evaluate ∫ (

𝟏+𝒄𝒐𝒔𝟐 𝒙 𝒄𝒐𝒔 𝒙

) 𝒅𝒙

Solution: Step 1: Integrate separately and add the result, giving us ∫(

1 + 𝑐𝑜𝑠2 𝑥 1 𝑐𝑜𝑠2 𝑥 ) 𝑑𝑥 = ∫ 𝑑𝑥 + ∫ 𝑑𝑥 cos 𝑥 cos 𝑥 cos 𝑥

Step 2: We know the fact that

1

cos

= sec and

𝑐𝑜𝑠 2𝑥 cos 𝑥

= 𝑐𝑜𝑠𝑥 , we have

1 𝑐𝑜𝑠 𝑥 𝑑𝑥 + ∫ 𝑑𝑥 = ∫ sec 𝑥 𝑑𝑥 + ∫ cos 𝑥 𝑑𝑥 cos 𝑥 cos 𝑥 Step 3: Use the formula ∫ 𝒔𝒆 𝒅𝒙 = 𝒍 𝒔𝒆 𝒙 + 𝒕𝒂 + 𝑪 and ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶, we have = 𝒍𝒏|𝐬𝐞𝐜 𝒙 + 𝐭𝐚𝐧 𝒙| + 𝐬𝐢𝐧 𝒙 + 𝑪 ∫

∫(

𝟏+𝒄𝒐𝒔𝟐 𝒙 ) 𝒅𝒙 𝒄𝒐𝒔 𝒙

2

= = 𝒍𝒏|𝐬𝐞𝐜 𝒙 + 𝐭𝐚𝐧 𝒙| + 𝐬𝐢𝐧 𝒙 + 𝑪

Explore

Here are some enrichment activities for you to work on to master and strengthen the basic concepts you have learned from this lesson. Assessment 1: Try this! Directions: Evaluate the following integrals below. Use a separate sheet of paper for your answers. Show your complete solution. 1. 2. 3. 4. 5. 6.

∫ 12 𝑑𝑥 ∫ 𝑥 5 𝑑𝑥 ∫ 6𝑥2 𝑑𝑥 ∫(3𝑥 + 7) 𝑑𝑥 ∫ 3𝑥 𝑑𝑥 ∫ 3𝑥+3 𝑑𝑥

7. ∫

23 𝑥

𝑑𝑥

8. ∫ 6 tan 𝑥 𝑑𝑥 9. ∫ −2 cos 𝑥 𝑑𝑥 10. ∫ 5 cot 𝑥 𝑑𝑥

Deepen Assessment 2: Let’s Solve! Directions: Evaluate the following integrals below. Use a separate sheet of paper for your answers. Show your complete solution. 1. ∫ 𝑥 −7 𝑑𝑥 𝑥

2. (∫ 𝑒 − 3.

1

) 𝑑𝑥

9𝑥 5𝑥 3 +3𝑥 2+𝑥 ∫ 𝑥2

𝑑𝑥

4. (5 tan 𝑥 − 4𝑐𝑠𝑐2 𝑥)𝑑𝑥 1

5. ∫ 𝑐𝑠𝑐2 𝑥 𝑑𝑥 6

Great job! You have understood the lesson.

Gauge Directions: Read carefully each item. Use a separate sheet for your answers. 1. What is defined as functions that cannot be written using the algebraical operations of addition, multiplication, or their inverse operations? A. Exponential functions C. Trigonometric functions B. Logarithmic functions D. Transcendental functions 2. The following are functions that can be best integrated using Power Rule, EXCEPT? 2

B. ∫ 𝑥 5 𝑑𝑥 C. ∫ 4 𝑑𝑥 D. ∫ 𝑥 −2 𝑑𝑥 A. ∫ √𝑥 2 𝑑𝑥 3. What is the anti-differentiation rule to be used in order to integrate the function 3

∫ √𝑥2 ? A. Constant rule B. Constant multiple rule 5

C. Power rule D. Sum rule

4. If the given integrand is √𝑥 5 , what would you do first in order to integrate the function? A. Apply the constant multiple rule to integrate the function. B. Apply the power rule to integrate the function. 3

3

C. Transform the integrand into its exponential form, 𝑥 5. 5

D. Transform the integrand into its exponential form, 𝑥 3. 5. Jacob aims to finish his homework in Basic Calculus. One of the problem in his homework is to evaluate ∫ 6 𝑡𝑎𝑛𝑥 𝑑𝑥. What would be the first step for him to solve the integral? A. ∫ 6𝑑𝑥 − ∫ 𝑡𝑎𝑛𝑥𝑑𝑥 C. 6 ∫ 𝑡𝑎𝑛𝑥𝑑𝑥 D. −6𝑙𝑛|𝑐𝑜𝑠𝑥|𝑑𝑥 B.∫ 6. −𝑙𝑛|𝑐𝑜𝑠𝑥|𝑑𝑥 For numbers 6 – 15. Evaluate the following integrals. 6. ∫ 𝑥 5 𝑑𝑥 7. ∫ √𝑥 𝑑𝑥

A. 5𝑥 + 𝐶 33

A. 2 √𝑥2 + 𝐶

8. ∫ √𝑥2 𝑑𝑥 5

A.

5

1

B. 5 𝑥+C

B.

2 3

57

3

+𝐶

B.

𝑥5 5

+𝐶

B. −𝑥 +

9. ∫(1 − 𝑥 4 )𝑑𝑥

A. 𝑥 +

10. ∫(3𝑥 + 10) 𝑑𝑥

A. 3𝑥2 + 10𝑥 + 𝐶

7

D.

3

D.

6

√𝑥2 + 𝐶

5𝑥 √𝑥2 7

1

C. 𝑥 + 𝐶

√𝑥 5 + 𝐶 𝑥5 5

+𝐶

B. 3𝑥2 − 10𝑥 + 𝐶

C. √𝑥2 + 𝐶 2

C.

2𝑥 √𝑥 3

+𝐶

77

7 5 7 √𝑥 5

+𝐶

D. √𝑥 5 + 𝐶

𝑥5

+𝐶

D. −𝑥 −

C. 𝑥 − C.

𝑋6 +𝐶 6

3𝑥 2 2

5

− 10𝑥 + 𝐶

5

D.

3𝑥 2 2

𝑥5 5

+𝐶

+ 10𝑥 + 𝐶

11. ∫ 12𝑥 𝑑𝑥 3

12. ∫ 𝑥 𝑑𝑥

1

A. 12𝑥 + 𝐶

B.

A. 3𝑙𝑛|𝑥| + 𝐶

B.3𝑙𝑛|𝑥| + 𝐶

12𝑥

+𝐶

13. ∫(1 − cos 𝑥)𝑑𝑥 A. 𝑥 − cos 𝑥 + 𝐶 B. 𝑥 − sin 𝑥 + 𝐶 14. ∫ tan 𝑥 sec 𝑥 𝑑𝑥 A. cos 𝑥 + 𝐶 B. sin 𝑥 + 𝐶 15. ∫ 6 cot 𝑥 𝑑𝑥 A. 6 ln|sin 𝑥 | + 𝐶 B.6 ln|cos 𝑥| + 𝐶

12𝑥

C. ln 12 + 𝐶 1

D.

ln 12 12𝑥

+𝐶

1

C. 3 𝑙𝑛|𝑥| + 𝐶

D. 𝑥𝑙𝑛|𝑥| + 𝐶

C. 𝑥 + cos 𝑥 + 𝐶

D. 𝑥 ∓ sin 𝑥 + 𝐶

C. csc 𝑥 + 𝐶

D. sec 𝑥 + 𝐶

C. 6 ln|csc 𝑥| + 𝐶

D. 6 ln|sec 𝑥| + 𝐶

3

References Printed Materials: Canlapan, R. B. (2017). DIWA Senior High School Series: Basic Calculus. Diwa Learning System Inc. Bacani, J. B., et al. (2016). Basic Calculus (For Senior High School). Books Atbp. Publishing Corp. Balmaceda, J. P., et al. (2016). Teaching Guide for Senior High School Basic Calculus. Commission on Higher Education Anton, H. et al. (2012). Calculus Single Variable. Wiley. Riley, K. et al. (2006). Mathematical Methods for Physics and Engineering. A Comprehensive Guide. Cambridge University Press

Website: Antiderivatives. Retrieved February 5, 2021 from https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus__Ea rly_Transcendentals_(Stewart)/04%3A_Applications_of_Differentiation/4.09 %3A_Antiderivatives Indefinite Integrals. Retrieved February 5, 2021 from https://tutorial.math.lamar.edu/classes/calci/indefiniteintegrals.aspx Antiderivatives and Indefinite Integrals. Retrieved February 5, 2021 from hhttps://www.khanacademy.org/math/ap-calculus-ab/ab-integrationnew/ab-67/e/antiderivativesttp://www.mathcentre.ac.uk/resources/uploaded/mcty-limits-2009-1.pdf...