5-7: Inverse Trig Functions, 2
Objectives:
1. To integrate
functions whose
antiderivative
involves an inverse
trig function
Assignment:
• P. 385-387: 1-45 eoo,
53, 55, 63, 67, 82
Warm Up 1
Solve 𝑥 2 − 6𝑥 + 3 = 0 by completing the
square.
Warm Up 2
Write 𝑦 = 3𝑥 2 + 3𝑥 + 3 in the form
𝑦 = 𝑎 𝑥 − ℎ 2 + 𝑘.
Objective 1
You will be able to integrate
functions whose antiderivative
involves an inverse trig function
Exercise 1
Find the derivative of 𝑦 = arcsin
is a function of 𝑥.
𝑢
𝑎
, where 𝑢
Exercise 2
Using some clever algebraic manipulation
and an unconventional substitution, find
𝑑𝑢
where 𝑢 is a function of 𝑥 and 𝑎 > 0.
2
2
𝑎 −𝑢
Exercise 3
Prove that
𝑑
𝑑𝑥
𝑑𝑢
𝑎2 +𝑢2
that
arctan 𝑢
of 𝑥 and 𝑎 > 0.
1
𝑢
= arctan
+𝐶
𝑎
𝑎
𝑢′
=
where 𝑢 is a
2
1+𝑢
given
function
What’s Your Sine?
Notice that the derivatives of arcsin 𝑥 and
arccos 𝑥 only differ by a sign.
𝑑
𝑢′
arcsin 𝑢 =
𝑑𝑥
1 − 𝑢2
𝑑
𝑢′
arccos 𝑢 = −
𝑑𝑥
1 − 𝑢2
As such, when writing integration rules, we’ll
only concern ourselves with arcsin 𝑥.
Inverse Trig Integrals
Let 𝑢 be a differentiable function of 𝑥 with 𝑎 > 0.
1
𝑢
arctan
+𝐶
𝑎
𝑎
𝑑𝑢
𝑎2
−
𝑢2
𝑑𝑢
=
𝑢
𝑢2
−
𝑎2
=
𝑑𝑢
=
𝑎2 + 𝑢2
𝑢
arcsin
+𝐶
𝑎
1
arcsec
𝑎
𝑢
𝑎
+𝐶
Exercise 4
Integrate.
1.
𝑑𝑥
4−𝑥 2
2.
𝑑𝑥
2+9𝑥 2
3.
𝑑𝑥
𝑥 4𝑥 2 −9
Exercise 5
Find
𝑑𝑥
.
𝑒 2𝑥 −1
Exercise 6
Find
𝑥+2
4−𝑥 2
𝑑𝑥.
Occasionally, you
have to split up the
integrand to get it to
match up with a
particular integration
rule.
Completing the Square
Because integrals involving inverse trig functions
involve the sum or difference of squares,
completing the square is a useful technique to
integrate some of these expressions.
Exercise 7
Find
𝑑𝑥
.
2
𝑥 −4𝑥+7
Exercise 8
Find the area of the
region bounded by the
1
graph of 𝑦 =
,
2
3𝑥−𝑥
the 𝑥-axis and the lines
𝑥 = 3/2 and 𝑥 = 9/4.
Advanced Techniques to
Come!
Integration by parts
Trig substitution
Partial Fractions
Improper integrals
Exercise 9
Integrate.
1.
𝑑𝑥
𝑥 𝑥 2 −1
2.
𝑥 𝑑𝑥
𝑥 2 −1
3.
𝑑𝑥
𝑥 2 −1
Exercise 10
Integrate.
1.
𝑑𝑥
𝑥 ln 𝑥
2.
ln 𝑥
𝑑𝑥
𝑥
3.
ln 𝑥 𝑑𝑥
5-7: Inverse Trig Functions, 2
Objectives:
1. To integrate
functions whose
antiderivative
involves an inverse
trig function
Assignment:
• P. 385-387: 1-45 eoo,
53, 55, 63, 67, 82