Calculus- Early Transcendentals, 2021a

6.3. Indefinite Integrals 243
Exercise 6.2.8 Find the derivative of G(x)=
Exercise 6.2.9 Find the derivative of G(x)=
Exercise 6.2.10 Find the derivative of G(x)=
Exercise 6.2.11 Find the derivative of G(x)=
Exercise 6.2.12 Find the derivative of G(x)=
∫ x 2
1
∫ x
1
∫ x 2
1
∫ x
1
∫ x 2
t 2 − 3tdt
e t2 dt
10x
e t2 dt
tan(t 2 )dt
tan(t 2 )dt
Exercise 6.2.13 Suppose ∫ 4
1 f (x)dx = 2 and ∫ 4
1 g(x)dx = 7.
2 f (x))dx.
Find ∫ 4
1 (5 f (x)+3g(x))dx and ∫ 4
1 (6 −
Exercise 6.2.14 Suppose ∫ 5
−2 f (x)dx = 3 and ∫ 5
1 f (x)dx = −2. Find ∫ 1
−2 f (x)dx.
Exercise 6.2.15 If f is continuous on [a,b], we define the average of f (x) on [a,b] to be
avg [a,b] ( f )= 1 ∫ b
f (x)dx
b − a a
(a) What is the average of √ x on the interval [0,1]?
(b) If the average of f (x) on [0,2] and on [2,5] are 6 and 4 respectively, then what is its average on
[0,5]?
6.3 Indefinite Integrals
In this section we focus on computing indefinite integrals. The process of finding the indefinite integral is
called integration (or integrating f (x)).
Example 6.21: Indefinite Integral
Evaluate the following indefinite integral:
∫
x 5 + 3x − 2dx.
Solution. Since this is asking for the most general anti-derivative we have:
∫
x 5 + 3x − 2dx = x6
6 + 3x2 − 2x +C
2