5.4 Indefinite Integrals and the Net Change Theorem
1. Definition (Indefinite integral) If f (x) is a continuous function, the general form of the antiderivatives of f is called
the indefinite integral of f and it is denoted by
Z
f (x)dx
Notice the lack of lower and upper bounds compared to a definite integral.
2. Formulas We redo the formulas for antiderivatives in terms of indefinite integrals :
• (Linear property of an integral) For any constants α, β, and continuous functions f (x) and g(x),
Z
Z
Z
αf (x) + βg(x)dx = α f (x)dx + β g(x)dx
Z
•
1dx = x + C
Z
•
xn dx =
xn+1
+ C (n 6= −1)
n+1
 Z


sin xdx



 Z
cos xdx
•

Z





sec2 xdx
(Summary)
Integrals
 Z


sec x tan xdx



 Z
csc x cot xdx
•

Z





csc2 xdx
= − cos x + C
=
sin x + C
=
tan x + C






Indefinite integral
:





Definite integral
:
=
sec x + C
=
− csc x + C
= − cot x + C
Z
f (x)dx
Z
:
anti-derivatives of f (x)
:
the area under the graph y = f (x)
on [a, b].
b
f (x)dx
a
Fundamental Theorem
Indefinite integral
Definite integral
of Calculus
Z
Z
a
3. Example Find the indefinite integral :
Z
cos θ
dθ
sin2 θ
Z
1
+ 3 − csc2 xdx
x2
(a)
(b)
Answers : (a) − csc θ + C,
(b)−
b
f (x)dx = F (b) − F (a)
f (x)dx = F (x) + C
1
+ 3x + cot x + C
x
1
4. Example Evaluate the indefinite integral and interpret the geometric meaning :
Z
4
(a)
√
t(1 + t)dt
1
Z
4
r
(b)
1
Z
π/3
(c)
−π/6
Z
5
dx
x
sin θ + sin θ tan2 θ
dθ
sec2 θ
3π/2
| sin x|dx
(d)
0
Answers : (a)
256
15
√
(b) 2 5
(c)
−1 +
2
√
3
(d) 3
5. Theorem (The Net Change Theorem) When a changing quantity Q(t) changes by a rate r(t), the net change of the
quantity during a ≤ t ≤ b is
Z
Q(b) − Q(a) =
b
r(t)dt
a
100
Z
6. Example
(a) If a particle moves on x−axis with the velocity v(t)unit/sec,
v(t)dt represents displacement of
0
the particle after 100 seconds.
100
Z
|v(t)|dt represents the total travel
(b) If a particle moves on x−axis with the velocity v(t)unit/sec,
0
distance during the first 100 seconds.
Z
(c) If water leaks from a water tank by a rate r(t) gal/min,
60
r(t)dt represents the total water loss
0
during the first 60minutes.
Z
20
(d) If ρ(x)(gram/cm) is the density of a rod at the point x cm from the left end,
the mass of a portion of the rod between 10cm and 20cm from left end.
ρ(x)dx represents
10
(e) If b(t)(persons/month) is the number of births per month(called birth rate) on earth and
d(t)(persons/month) is the number of deaths per month(called death rate) on earth,
Z 12
b(t) − d(t)dt represents the total change of world population during the first 12 months since
0
the observation.
2
7. Example
(a) The acceleration of a particle moving along x−axis is
a(t) = 2t + 3 (unit/sec2 )
When the initial velocity v(0) = −4, find the total travel distance during 0 ≤ t ≤ 2.
(b) Oil leaks from the bottom of a container at a rate of
√
r(t) = 10 − t (liters/min)
Find the total amount of oil loss during the first 25 minutes.
2
Z
Answers :
2
Z
1
0
0
2
Z
2
−(t + 3t − 4)dt +
t + 3t − 4 dt =
(a)
1
Figure for (a)
3
2
t + 3t − 4dt = 5
25
Z
10 −
(b)
0
√
tdt =
500
liters
3