5-2: Natural Logs, 2
Objectives:
Assignment:
1. To integrate rational
functions using the Log
Rule
• P. 338-340: 1-27 odd, 37,
43, 47-57 odd, 67, 87, 89,
91, 97-100
• P. 338-340: 29-35 odd,
2. To integrate trig functions
39, 46, 53, 69, 90
Warm Up
Which of the following do not belong?
6𝑥
𝑑𝑥
3𝑥 2 + 1
6𝑥
3𝑥 2
6𝑥
3𝑥 2 + 1
2 𝑑𝑥
+1
𝑑𝑥
6𝑥 3𝑥 2 + 1 𝑑𝑥
Warm Up
Which of the following do not belong?
6𝑥
𝑑𝑥
3𝑥 2 + 1
Cannot be integrated using a
combination of 𝑢-substitution
and the General Power Rule
Objective 1
You will be able to integrate
rational functions using the Log
Rule
Exercise 1
Find the derivative of 𝑦 = ln 2𝑥 + 3 .
𝑢>0
𝑦 = ln 2𝑥 + 3
2
𝑦′ =
2𝑥 + 3
𝑑
1 𝑑𝑢 𝑢′
ln 𝑢 = ∙
=
𝑑𝑥
𝑢 𝑑𝑥 𝑢
This is a
rational function
𝑢′
𝑢
Exercise 2
Find the antiderivative
2
𝑑𝑥
2𝑥 + 3
2
𝑑𝑥.
2𝑥+3
𝑥
ln 𝑥 =
1
1
𝑑𝑡
𝑡
ln 2𝑥 + 3 + 𝐶
𝑥>0
ln 2𝑥 + 3 + 𝐶
Log Rule for Integration
Let 𝑢 be a differentiable function of 𝑥.
1
𝑑𝑥 = ln 𝑥 + 𝐶
𝑥
𝑢′
𝑑𝑥 = ln 𝑢 + 𝐶
𝑢
1
𝑑𝑢 = ln 𝑢 + 𝐶
𝑢
Exercise 3
Find
2
𝑑𝑥.
𝑥
Exercise 4
Find
1
𝑑𝑥.
4𝑥−1
Exercise 5
Find the area the
region bounded by
the graph of
𝑥
𝑦 = 2 , the
𝑥 +1
𝑥-axis, and the line
𝑥 = 3.
Exercise 6
Integrate the following.
1.
3𝑥 2 +1
𝑑𝑥
𝑥 3 +𝑥
2.
sec2 𝑥
𝑑𝑥
tan 𝑥
Exercise 7
Integrate the following.
3.
𝑥+1
𝑑𝑥
𝑥 2 +2𝑥
4.
1
𝑑𝑥
3𝑥+2
Exercise 8
Which of the following does not belong?
2𝑥
𝑑𝑥
2
𝑥 +1
𝑥2 + 𝑥 + 1
𝑑𝑥
2
𝑥 +1
𝑥
𝑑𝑥
2
𝑥 +1
𝑥+1
𝑑𝑥
2
𝑥 + 2𝑥
Long Division
If the degree of the
numerator is greater
than or equal to the
degree of
denominator, rewrite
the integrand by
dividing.
𝑥2 + 𝑥 + 1
𝑑𝑥
2
𝑥 +1
𝑥
1+ 2
𝑑𝑥
𝑥 +1
Exercise 9
Find
𝑥 2 +𝑥+1
𝑑𝑥.
2
𝑥 +1
Exercise 10
Find
2𝑥
𝑑𝑥.
2
𝑥+1
Exercise 11
Solve the differential equation
𝑑𝑦
𝑑𝑥
=
1
.
𝑥 ln 𝑥
Objective 2
You will be able to integrate
trig functions
Exercise 12
Find
tan 𝑥 𝑑𝑥.
Exercise 13
Find sec 𝑥 𝑑𝑥.
Trig Rules
sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶
cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶
tan 𝑢 𝑑𝑢 = − ln cos 𝑢 + 𝐶
cot 𝑢 𝑑𝑢 = ln sin 𝑢 + 𝐶
sec 𝑢 𝑑𝑢 = ln sec 𝑢 + tan 𝑢 + 𝐶
csc 𝑢 𝑑𝑢 = − ln csc 𝑢 + cot 𝑢 + 𝐶
Exercise 14
Evaluate
𝜋/4
0
1 + tan2 𝑥 𝑑𝑥.
Exercise 15
Find the average value of 𝑦 = tan 𝑥 on
𝜋
0,
4
.
5-2: Natural Logs, 2
Objectives:
Assignment:
1. To integrate rational
functions using the Log
Rule
• P. 338-340: 1-27 odd, 37,
43, 47-57 odd, 67, 87, 89,
91, 97-100
• P. 338-340: 29-35 odd,
2. To integrate trig functions
39, 46, 53, 69, 90