Integrals Related to Inverse Trig, Inverse Hyperbolic Functions Lesson 9.5 Review • Recall derivatives of inverse trig functions d 1 du 1 sin u , u 1 dx 1
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Transcript Integrals Related to Inverse Trig, Inverse Hyperbolic Functions Lesson 9.5 Review • Recall derivatives of inverse trig functions d 1 du 1 sin u , u 1 dx 1
Integrals Related to Inverse Trig,
Inverse Hyperbolic Functions
Lesson 9.5
Review
• Recall derivatives of inverse trig functions
d
1 du
1
sin u
, u 1
dx
1 u 2 dx
d
1 du
1
tan u
2
dx
1 u dx
d
1
du
1
sec u
, u 1
dx
u u 2 1 dx
2
Integrals Using Same
Relationships
du
u
a 2 u 2 arcsin a C
du
1
u
a 2 u 2 a arctan a C
du
1
u
u u 2 a 2 a arcsec a C
When given
integral problems,
look for these
patterns
3
Identifying Patterns
• For each of the integrals below, which
inverse trig function is involved?
4dx
13 16 x 2
dx
9 x
2
x
dx
25 x 2 4
dx
x 2 2 x 10
Hint: use
completing
the square
4
Warning
• Many integrals look like the inverse trig
forms
• Which of the following are of the inverse
trig forms?
x dx
1 x2
dx
1 x2
x dx
1 x2
dx
1 x
2
If they are not,
how are they
integrated?
5
Try These
• Look for the pattern or how the expression
can be manipulated into one of the
patterns
8dx
1 16 x 2
x dx
1 25 x 2
dx
4 x 2 4 x 15
x 5
x 10 x 16
2
dx
6
Integrals Involving Inverse
Hyperbolic Functions
1
u
du sinh
C
a
u 2 a2
1
1 u
du cosh
C
a
u2 a2
1
1
1
-1 u
a 2 u 2 du a tanh a C
1
1
1 u
u a 2 u 2 du a sech a C
7
Try It!
4
• Note the definite integral
1
1
9 4 x2
dx
• What is the a, the u, the du?
a = 3, u = 2x, du = 2 dx
4
1
2
dx
2 1 9 4 x2
4
1
1 2 x
sinh
2
3 1
8
Application
• Find the area enclosed by
x = -¼, x = ¼, y = 0, and
y
1
1 4 x2
• Which pattern does this match?
• What is the a, the u, the du?
9
Assignment
• Lesson 9.5
• Page 381
• Exercises 1 – 35 odd
10