c.7.2 - indefinite integrals

04/19/2304/19/23 Calculus - SantowskiCalculus - Santowski 11

C.7.2 - Indefinite C.7.2 - Indefinite IntegralsIntegrals

Calculus - SantowskiCalculus - Santowski


Lesson ObjectivesLesson Objectives

1. Define an indefinite integral 2. Recognize the role of and determine the value of a

constant of integration 3. Understand the notation of f(x)dx 4. Learn several basic properties of integrals 5. Integrate basic functions like power, exponential,

simple trigonometric functions 6. Apply concepts of indefinite integrals to a real

world problems


Fast FiveFast Five


(A) Review - Antiderivatives(A) Review - Antiderivatives

Recall that working with antiderivatives was Recall that working with antiderivatives was simply our way of “working backwards” simply our way of “working backwards”

In determining antiderivatives, we were simply In determining antiderivatives, we were simply looking to find out what equation we started looking to find out what equation we started with in order to produce the derivative that with in order to produce the derivative that was before uswas before us

Ex. Find the antiderivative of a(t) = 3t - 6eEx. Find the antiderivative of a(t) = 3t - 6e2t2t


(B) Indefinite Integrals - Definitions(B) Indefinite Integrals - Definitions

Definitions: an anti-derivative of Definitions: an anti-derivative of f(x)f(x) is any function is any function F(x)F(x) such such that that F`(x)F`(x) = = f(xf(x) ) If If F(x)F(x) is any anti-derivative of is any anti-derivative of f(x)f(x) then the then the most general anti-derivative of most general anti-derivative of f(x)f(x) is called is called an indefinite an indefinite integralintegral and denoted and denoted f(x)dxf(x)dx = = F(x)F(x) + + C C where where CC is any is any constantconstant

In this definition the In this definition the is called the integral symbol, is called the integral symbol, f(x)f(x) is is called the integrand, called the integrand, xx is called the integration variable and the is called the integration variable and the ““CC” is called the constant of integration ” is called the constant of integration So we can interpret So we can interpret the statement the statement f(x)dxf(x)dx as “determine the integral of f(x) with as “determine the integral of f(x) with respect to x”respect to x”

The process of finding an indefinite integral (or simply an The process of finding an indefinite integral (or simply an integral) is called integrationintegral) is called integration


(C) Review - Common Integrals(C) Review - Common Integrals

Here is a list of common integrals:Here is a list of common integrals:

€

k( )dx =∫ k × dx∫ = kx + C

x n( ) ∫ dx =

x n +1

n +1+ C

ekx( ) dx =

1

kekx∫ + C

akx( ) ∫ dx =

1

k lnaakx + C

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟ dx∫ = ln x + C

sin kx( ) dx = −1

kcos kx( )∫ + C

cos kx( ) dx =1

k∫ sin kx( ) + C


(D) Properties of Indefinite Integrals(D) Properties of Indefinite Integrals

Constant Multiple rule:Constant Multiple rule: [c [c f(x)]dx = c f(x)]dx = c f(x)dx and f(x)dx and -f(x)dx = - -f(x)dx = - f(x)dxf(x)dx

Sum and Difference Rule:Sum and Difference Rule: [f(x) [f(x) ++ g(x)]dx = g(x)]dx = f(x)dx f(x)dx ++ g(x)dxg(x)dx

which is similar to rules we have seen for derivativeswhich is similar to rules we have seen for derivatives


(D) Properties of Indefinite Integrals(D) Properties of Indefinite Integrals

And two other interesting “properties” need to be And two other interesting “properties” need to be highlighted:highlighted:

Interpret what the following 2 statement mean:Interpret what the following 2 statement mean:

g`(x)dx = g(x) + Cg`(x)dx = g(x) + C

d/dx d/dx f(x)dx = f(x)f(x)dx = f(x)


(E) Examples(E) Examples

(x(x44 + 3x – 9)dx = + 3x – 9)dx = xx44dx + 3 dx + 3 xdx - 9 xdx - 9 dxdx (x(x44 + 3x – 9)dx = 1/5 x + 3x – 9)dx = 1/5 x55 + 3/2 x + 3/2 x22 – 9x + C – 9x + C

ee2x2xdx = dx = sin(2x)dx = sin(2x)dx = (x(x22x)dx = x)dx = (cos(cos + 2sin3 + 2sin3)d)d = = (8x + sec(8x + sec22x)dx = x)dx = (2 - (2 - x)x)22dx = dx =


(F) Examples(F) Examples

Continue now with these questions on lineContinue now with these questions on line Problems & Solutions with Antiderivatives / Problems & Solutions with Antiderivatives /

Indefinite Integrals from Visual CalculusIndefinite Integrals from Visual Calculus


(G) Indefinite Integrals with Initial (G) Indefinite Integrals with Initial ConditionsConditions

Given that Given that f(x)dx = F(x) + C, we can determine a f(x)dx = F(x) + C, we can determine a specific function if we knew what C was equal to specific function if we knew what C was equal to so if we knew something about the function F(x), so if we knew something about the function F(x), then we could solve for Cthen we could solve for C

Ex. Evaluate Ex. Evaluate (x(x33 – 3x + 1)dx if F(0) = -2 – 3x + 1)dx if F(0) = -2 F(x) = F(x) = xx33dx - 3 dx - 3 xdx + xdx + dx = ¼xdx = ¼x44 – 3/2x – 3/2x22 + x + C + x + C Since F(0) = -2 = ¼(0)Since F(0) = -2 = ¼(0)44 – 3/2(0) – 3/2(0)22 + (0) + C + (0) + C So C = -2 and So C = -2 and F(x) = ¼xF(x) = ¼x44 – 3/2x – 3/2x22 + x - 2 + x - 2


(H) Examples – Indefinite Integrals (H) Examples – Indefinite Integrals with Initial Conditionswith Initial Conditions

Problems & Solutions with Antiderivatives / Problems & Solutions with Antiderivatives / Indefinite Integrals and Initial Conditions from Indefinite Integrals and Initial Conditions from Visual CalculusVisual Calculus

Motion Problem #1 with Antiderivatives / Motion Problem #1 with Antiderivatives / Indefinite Integrals from Visual CalculusIndefinite Integrals from Visual Calculus

Motion Problem #2 with Antiderivatives / Motion Problem #2 with Antiderivatives / Indefinite Integrals from Visual CalculusIndefinite Integrals from Visual Calculus


(I) Internet Links(I) Internet Links

Calculus I (Math 2413) - Integrals from Paul DCalculus I (Math 2413) - Integrals from Paul Dawkinsawkins

Tutorial: The Indefinite Integral from Stefan Tutorial: The Indefinite Integral from Stefan Waner's site "Everything for Calculus”Waner's site "Everything for Calculus”

The Indefinite Integral from PK Ving's The Indefinite Integral from PK Ving's Problems & Solutions for Calculus 1Problems & Solutions for Calculus 1

Karl's Calculus Tutor - Integration Using Your Karl's Calculus Tutor - Integration Using Your Rear View MirrorRear View Mirror


(J) Homework(J) Homework

Textbook, p392-394Textbook, p392-394

(1) Algebra Practice: Q5-40 as needed + (1) Algebra Practice: Q5-40 as needed + varietyvariety

(2) Word problems: Q45-56 (economics)(2) Word problems: Q45-56 (economics) (3) Word problems: Q65-70 (motion)(3) Word problems: Q65-70 (motion)

c.7.2 - indefinite integrals

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