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Calculus - Santowski
04/21/23Calculus - Santowski 1
C.7.2 - Indefinite Integrals
Lesson Objectives
04/21/23Calculus - Santowski2
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
04/21/23Calculus - Santowski3
(A) Review - Antiderivatives
04/21/23Calculus - Santowski4
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
04/21/23Calculus - Santowski5
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
04/21/23Calculus - Santowski6
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
04/21/23Calculus - Santowski9
(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
04/21/23Calculus - Santowski10
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
04/21/23Calculus - Santowski12
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
04/21/23Calculus - Santowski13
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
04/21/23Calculus - Santowski14
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
04/21/23Calculus - Santowski15
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)
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