Transcript pwrpt 7.2

7.2
Trigonometric Integrals
Copyright © Cengage Learning. All rights reserved.
Trigonometric Identities
From Pre-Calc, there are trigonometric
identities which will be important in
finding some trig integrals:
or sin²x = 1- cos²x
sinxcosx = ½ (sin2x)
Expect a Quiz!
2
Trigonometric Integrals
In this section we use trigonometric identities to integrate
certain combinations of trigonometric functions.
We start with powers of sine and cosine.
Things to try:
If both Sin & Cos even: Try to substitute with Pythagorean or ½ trig identity
If both Sin & Cos odd: Try making one even & use Pythagorean trig identity for even one
If Sin is odd: Try making sin even, use sin²= (1- cos²) & then u-substitution
If Cos is odd: Try making cos even, use cos²= (1- sin²) & then u-substitution
3
Tips---MOST IMPORTANT!!!
Starting Goal for Finding Trig Integrals:
1.) Reduce the integral so that all trig
functions are not raised to a power.
2.) If a trig function is raised to a
power, find a way to get it's derivative
in the integrand so you can use usubstitution
4
Example 1
Find ∫ cos³x dx
Rewrite as: ∫ cos²x cosx dx
Substitute trig identity: ∫(1-sin²x)cosx dx
Use U-substitution: ∫(1-u²)du
Integrate: ∫(1-u²)du = u – 1/3 u³ + C
Substitute sin x back in: sinx – 1/3 sin³ x + C
5
Example 2
Find ∫ sin5x cos2x dx.
Solution:
We could convert cos2x to 1 – sin2x, but we would be left
with an expression in terms of sin x with no extra cos x
factor.
Instead, we separate a single sine factor and rewrite the
remaining sin4x factor in terms of cos x:
sin5 x cos2x = (sin2x)2 cos2x sin x
= (1 – cos2x)2 cos2x sin x
6
Example 2 – Solution
cont’d
Substituting u = cos x, we have du = –sin x dx and so
∫ sin5x cos2x dx = ∫ (sin2x)2 cos2x sin x dx
= ∫ (1 – cos2x)2 cos2x sin x dx
= ∫ (1 – u2)2 u2 (–du) = –∫ (u2 – 2u4 + u6)du
=
= – cos3x + cos5x – cos7x + C
7
Example 3
Evaluate
Solution:
If we write sin2x = 1 – cos2x, the integral is no simpler to
evaluate. Using the half-angle formula for sin2x, however,
we have
8
Example 3 – Solution
cont’d
Notice that we mentally made the substitution u = 2x when
integrating cos 2x.
9
Trigonometric Integrals
To summarize, we list guidelines to follow when evaluating
integrals of the form ∫ sinmx cosnx dx, where m  0 and
n  0 are integers.
10
Trigonometric Integrals
11
Trigonometric Integrals
Video Examples:
Sin,cos even: https://www.youtube.com/watch?v=rpbr2nH7lNY
Sin is odd:
https://www.youtube.com/watch?v=WYhyq_mTCZs
Cos is odd:
https://www.youtube.com/watch?v=RRDiT-djQPk
5 examples: https://www.youtube.com/watch?v=kw20LXdGLQc
30 minutes long, but good !
12
Homework:
Page 500 # 1-8 all, 11, 12, 15, 16
13
Part Two:
Tangent and Secant
&
Sin(mx) and Cox(nx)
14
Trig Identities:
Tan² x = Sec²x -1
Sec²x = 1 + Tan² x
15
Trigonometric Integrals
We can use a similar strategy to evaluate integrals of the
form ∫ tanmx secnx dx.
If sec is even: we can separate a sec2x factor and convert
the remaining (even) power of secant to an expression
involving tangent using the identity sec2x = 1 + tan2x.
Or, if tan is odd: we can separate a sec x tan x factor and
convert the remaining (even) power of tangent to secant.
Or, if there is no sec: we can make one tan² = sec²x -1
16
Example 5
Evaluate ∫ tan6x sec4x dx.
Solution:
If we separate one sec2x factor, we can express the
remaining sec2x factor in terms of tangent using the identity
sec2x = 1 + tan2x.
We can then evaluate the integral by substituting u = tan x
so that du = sec2x dx:
∫ tan6x sec4x dx = ∫ tan6x sec2x sec2x dx
17
Example 5 – Solution
cont’d
= ∫ tan6x (1 + tan2x) sec2x dx
= ∫ u6(1 + u2)du = ∫ (u6 + u8)du
=
= tan7x +
tan9x + C
18
Example 6
19
Trigonometric Integrals
The preceding examples demonstrate strategies for evaluating
integrals of the form ∫ tanmx secnx dx for two cases, which we
summarize here.
20
Trigonometric Integrals
For other cases, the guidelines are not as clear-cut. We
may need to use identities, integration by parts, and
occasionally a little ingenuity.
We will sometimes need to be able to integrate tan x by
using the formula given below:
Memorize!
21
Trigonometric Integrals
We will also need the indefinite integral of secant:
Memorize!
22
Example 7
Find ∫ tan3x dx.
Solution:
Here only tan x occurs, so we use tan2x = sec2x – 1 to
rewrite a tan2x factor in terms of sec2x:
∫ tan3x dx = ∫ tan x tan2x dx
= ∫ tan x (sec2x – 1) dx
= ∫ tan x sec2x dx – ∫ tan x dx
23
Example 7 – Solution
=
cont’d
– ln |sec x| + C
In the first integral we mentally substituted u = tan x so that
du = sec2x dx.
24
Trigonometric Integrals
Finally, we can make use of another set of trigonometric
identities:
Memorize!
25
Example 9
Evaluate ∫ sin 4x cos 5x dx.
Solution:
This integral could be evaluated using integration by parts,
but it’s easier to use the identity in Equation 2(a) as follows:
∫ sin 4x cos 5x dx = ∫ [sin(–x) + sin 9x] dx
= ∫ (–sin x + sin 9x) dx
= (cos x – cos 9x) + C
26
Video Examples:
Tangent and Secant Problems:
https://www.youtube.com/watch?v=RGaDMqhOg8Y
https://www.youtube.com/watch?v=o8sIHlS17qc
5 examples: https://www.youtube.com/watch?v=YsVsuhdGRJk
30 minutes but good!
27
Homework:
Pge 501 # 22-32 even, 41,42,43
28