Lesson 5-4a
Indefinite Integrals Review
Icebreaker
Problem 1:
2
3 - 9) dx
(
3x
∫
= ¾ x4 – 9x + c
1
= ¾ 24 – 9(2) + c – (¾ 14 – 9(1) + c) =
Find the derivative of the following:
3x
∫x² (2t - 7) dt
= t² - 7t + c
= (2t – 7) dt  [2(3x) – 7](3)
= (2t – 7) dt  [2(x²) – 7](2x)
Objectives
• Solve indefinite integrals of algebraic,
exponential, logarithmic, and trigonometric
functions
• Understand the Net Change Theorem
• Use integrals to solve distance problems to
find the displacement or total distance traveled
Vocabulary
• Indefinite Integral – is a function or a family
of functions
• Distance – the total distance traveled by an
object between two points in time
• Displacement – the net change in position
between two points in time
Basic Differentiation Rules
d
---- (c) = 0
dx
Constant
d
---- (xⁿ) = nxn-1
dx
Power Rule
d
d
---- [cf(x)] = c ---- f(x)
dx
dx
Constant Multiple Rule
d
---- (ex) = ex
dx
Natural Exponent
d
1
---- (ln x) = ----dx
x
Natural Logarithms
Trigonometric Functions
Differentiation Rules
d
---- (sin x) = cos x
dx
d
---- (cos x) = –sin x
dx
d
---- (tan x) = sec² x
dx
d
---- (cot x) = –csc² x
dx
d
---- (sec x) = sec x • tan x
dx
d
---- (csc x) = –csc x • cot x
dx
Hint: The derivative of trig functions (the “co-functions”) that begin
with a “c” are negative.
Derivatives of Inverse
Trigonometric Functions
d
1
-1
---- (sin x) = -----------dx
√1 - x²
d
-1
-1
---- (cos x) = ----------dx
√1 - x²
d
1
-1
---- (tan x) = ------------dx
1 + x²
d
-1
-1
---- (cot x) = ------------dx
1 + x²
d
1
-1
---- (sec x) = -------------dx
x √ x² - 1
d
-1
-1
---- (csc x) = ------------dx
x √ x² - 1
Interesting Note:
If f is any one-to-one differentiable function, it can be proved that its inverse
function f-1 is also differentiable, except where its tangents are vertical.
Other Differentiation Rules
Constant to Variable Exponent Rule
d
----- [ax] = ax ln a
dx
This is a simple example of logarithmic differentiation
that we will examine in a later problem.
Sum and Difference Rules
d
d
d
---- [f(x) +/- g(x)] = ---- f(x) +/- ---- g(x)
dx
dx
dx
In words: the derivative can be applied across an
addition or subtraction. This is not true for a
multiplication or a division as the next two rules
demonstrate.
Indefinite Integration Review
1.
∫
2.
∫a
3.
∫
ex dx
x
dx
=
4.
∫ cos(x) dx
=
5.
6.
sin(x) dx
=
=
∫
sec2(x) dx
=
∫
csc2(x) dx
=
Indefinite Integration Review
7.
8.
9.
∫ sec(x)tan(x) dx
∫ csc(x)cot(x) dx
∫
1
---------- dx
x² + 1
=
=
=
∫
1
---------- dx
1 - x²
11.
∫
1
----- dx
x
12.
∫
xn dx
10.
=
=
=
Example Problems with TI-89
Evaluating indefinite integral with our calculator:
∫
(-x2 + 4x – 3)dx
=
-⅓x3 + 2x2 – 3x + C
Hit F3 select integration; type in function (t²), integrate
with respect to (t), lower limit of integration (1),
upper limit of integration (x); close ). Type , and
differentiate with respect to x and close ). Should
look like this:
∫(-x^2 + 4x – 3,x,)
C is missing from calculator answer
More Practice Problems
Now use your knowledge of the formulas and integration
rules to evaluate the following:
1.
∫
(-x2 + 4x – 3)dx
2.
∫
(2x – 1)2 dx
=
∫
3
(----- - 1) dx
x²
=
3.
=
More Practice Problems
Now use your knowledge of the formulas and integration
rules to evaluate the following:
1
(x - -----) dx
x
4.
∫
5.
∫ (3sec(x)tan(x) – 2csc(x)cot(x)) dx
6.
∫
=
(2sec2(x) + 4csc2(x)) dx
=
=
More Practice Problems
Now use your knowledge of the formulas and integration
rules to evaluate the following:
7.
8.
9.
∫
1
(1 - ---------) dx
x² + 1
=
∫
3
(-----------) dx
1- x²
∫
4 – x - x²
(---------------) dx
2x
=
=
More Practice Problems
Now use your knowledge of the formulas and integration
rules to evaluate the following:
10.
∫
1 - sin²(x)
(---------------) dx
cos²(x)
=
Summary & Homework
• Summary:
– Definite Integrals are a number
– Evaluated at endpoints of integration
– Indefinite Integrals are antiderivatives
• Homework:
– Day One: pg 411-413: 1, 7, 8, 17, 20, 23, 33,
– Day Two: pg 411-413: 59, 62