Calculus - Santowski

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C.7.2 - Indefinite Integrals

Lesson Objectives

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1. Define an indefinite integral2. Recognize the role of and determine

the value of a constant of integration3. Understand the notation of f(x)dx4. Learn several basic properties of

integrals5. Integrate basic functions like power,

exponential, simple trigonometric functions

6. Apply concepts of indefinite integrals to a real world problems

Fast Five

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(A) Review - Antiderivatives

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

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

Ex. Find the antiderivative of a(t) = 3t - 6e2t

(B) Indefinite Integrals - Definitions

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

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

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

(C) Review - Common Integrals

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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

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

f(x)dx = - f(x)dx

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

which is similar to rules we have seen for derivatives

(D) Properties of Indefinite Integrals

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

Interpret what the following 2 statement mean: Use your TI-89 to help you with these 2 questions

Let f(x) = x3 - 2x

What is the answer for f `(x)dx ….?

What is the answer for d/dx f(x)dx ….. ?

(E) Examples

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(x4 + 3x – 9)dx = x4dx + 3 xdx - 9 dx(x4 + 3x – 9)dx = 1/5 x5 + 3/2 x2 – 9x + C

e2xdx = sin(2x)dx = (x2x)dx = (cos + 2sin3)d = (8x + sec2x)dx = (2 - x)2dx =

(F) Examples

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Continue now with these questions on lineProblems & Solutions with Antiderivatives /

Indefinite Integrals from Visual Calculus

(G) Indefinite Integrals with Initial Conditions

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

Ex. Evaluate (x3 – 3x + 1)dx if F(0) = -2F(x) = x3dx - 3 xdx + dx = ¼x4 – 3/2x2 +

x + CSince F(0) = -2 = ¼(0)4 – 3/2(0)2 + (0) + CSo C = -2 and F(x) = ¼x4 – 3/2x2 + x - 2

(H) Examples – Indefinite Integrals with Initial Conditions

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

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

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

(I) Examples with Motion

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An object moves along a co-ordinate line with a velocity v(t) = 2 - 3t + t2 meters/sec. Its initial position is 2 m to the right of the origin.

(a) Determine the position of the object 4 seconds later

(b) Determine the total distance traveled in the first 4 seconds

(J) Examples – “B” Levels

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Sometimes, the product rule for differentiation can be used to find an antiderivative that is not obvious by inspection

So, by differentiating y = xlnx, find an antiderivative for y = lnx

Repeat for y = xex and y = xsinx

(K) Internet Links

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Calculus I (Math 2413) - Integrals from Paul Dawkins

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

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

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

(L) Homework

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Textbook, p392-394

(1) Algebra Practice: Q5-40 (AN+V)(2) Word problems: Q45-56 (economics)(3) Word problems: Q65-70 (motion)