5.4 Preparation
1. 5.4 The Fundamental Theorem of Calculus and Indefinite Integral
Once again, we begin by reviewing antiderivatives in section 4.9.
(1pt) Solve WebAssign 5.4 Preparation #1(4.9.013). Note that this is a 4.9 problem.
Because of the Fundamental Theorem of Calculus (in 5.3) we find it convenient to
use integral notation for the most general antiderivative. In particular
Definition 1.1. The most general
antiderivative of f (x) is also called the indefinite
R
integral of f and is denoted f (x) dx.
Z
0
In other words, if F (x) = f (x), then
f (x) dx = F (x) + C.
C is called the constant of integration.
(1pt) Use the above definition to solve WebAssign 5.4 Preparation #2(5.4conversionof4.9.013).
The next page is a table of integrals (antiderivatives) covered in the section on
antiderivatives using integral notation. Review and be sure to have memorized all of
these formulas.
5.4 Preparation End
1
5.4 Indefinite Integrals
(1)
R
k dx = kx + C
(2)
R
kg(x) dx = k
(3)
R
g(x) ± h(x) dx =
(4) n 6= −1,
R
R
g(x) dx
xn dx =
R
1
xn+1
n+1
(5)
R
x−1 dx = ln x + C
(6)
R
ex dx = ex + C
(7) a > 0 and a 6= 1 ,
R
g(x) ± h(x) dx
ax dx =
+C
1 x
a
ln a
(8)
R
sin x dx = − cos x + C
(9)
R
cos x dx = sin x + C
(10)
R
sec2 x dx = tan x + C
(11)
R
csc2 x dx = − cot x + C
(12)
R
sec x tan x dx = sec x + C
(13)
R
csc x cot x dx = − csc x + C
(14)
R
1
dx = arctan x + C
1 + x2
+C
2
5.4 Indefinite Integrals
3
2. Examples
Example 2.1. WebAssign 5.4 WebAssign (SCalcET 5.4.012)
Find the general inZ
4
dx
definite integral. (Use C for the constant of integration.)
5x2 + 4 + 2
x +1
Z
Example 2.2. WebAssign 5.4 WebAssign (SCalcET 5.4.508.XP) Integrate
5θ − 4 csc θ cot θ dθ
5.4 Indefinite Integrals
4
Z
Example 2.3. WebAssign 5.4 WebAssign (SCalcET 5.4.509.XP) Integrate
5(1 + tan2 α) dα
Exampler2.4. WebAssign 5.4 WebAssign (SCalcET 5.4.521.XP) Evaluate the inteZ 9
7
gral.
dx
x
1
5.4 Indefinite Integrals
5
Example
2.5. WebAssign 5.4 WebAssign (SCalcET 5.4.045) Evaluate the integral.
Z 1
(x − 8|x|) dx
−2
Example 2.6. WebAssign 5.4 WebAssign (SCalcET 5.4.059) The velocity function
(in meters per second) is given for a particle moving along a line.
v(t) = 3t − 8, 0 ≤ t ≤ 3.
(a) Find the displacement.
(b) Find the distance traveled by the particle during the given time interval.
R3
Note that displacement over the time interval [0, 3] is s(3) − s(0) = 0 v(t) dt, while
R3
R3
total distance traveled is |s(8/3) − s(0)| + |s(3) − s(8/3)| = 0 |v(t)| dt = 0 (speed) dt.
5.4 Indefinite Integrals
6
5.4 Homework
Drill:
(0pt) WebAssign Homework # 1(5.4.006), 2(5.4.009), 3(5.4.011)
(5pt) WebAssign Homework # 4(5.4.012), 5(5.4.016), 6(5.4.508.XP), 7(5.4.509.XP),
8(5.4.018)
(0pt) WebAssign Homework # 9(5.4.023), 10(5.4.027)
(2pt) WebAssign Homework # 11(5.4.028), 12(5.4.521.XP)
(0pt) WebAssign Homework # 13(5.4.031.MI), 14(5.4.033)
(1pt) WebAssign Homework # 15(5.4.035)
(0pt) WebAssign Homework # 16(5.4.037)
(3pt) WebAssign Homework # 17(5.4.038.MI), 18(5.4.045), 19(5.4.046)
Putting it together:
(1pt) WebAssign Homework # 20(5.4.059)