Chapter
7, Section 4
Trigonometric Substitution
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Trigonometric Substitutions (a
> 0)
For
integrals involving (a^2
- u^2)^(1/2)
, let u
= a sin ø
Then
(a^2
- u^2)^(1/2)
= a
cos ø,
where
( -pi/2 <=
ø <= pi/2
)
See chart on
page 506 for geometric drawing
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For
integrals involving (a^2
+ u^2)^(1/2)
, let u
= a tan ø
Then
(a^2
+ u^2)^(1/2)
= a
sec ø,
where
( -pi/2 <=
ø <= pi/2
)
See chart on page 506 for geometric drawing
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For
integrals involving (u^2
- a^2)^(1/2)
, let u
= a sec ø
Then
(u^2
- a^2)^(1/2)
= (+ or -)a
tan ø,
where
(0<= ø < pi/2
or pi/2 <= ø
<= pi )
Use
the positive value if u > a
and
Use
the negative value if u < -a.
See chart on page 506 for geometric drawing
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| Theorem 7.2 Special Integration
Formulas (a >0)
1.) ∫ sqrt(a^2
- u^2)du
= 1/2 * [a^2
arcsin (u/a) + u sqrt(a^2
- u^2)]
+ C
2.)
∫ sqrt(u^2
- a^2)du
= 1/2 * [u * sqrt(u^2
- a^2)
-
a^2
Ln| u +
sqrt(u^2
- a^2)|]
+ C
3.)
∫ sqrt(u^2
+ a^2)du
= 1/2 * [u * sqrt(u^2
+ a^2)
+
a^2
Ln| u +
sqrt(u^2
+ a^2)|]
+ C |
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Finding arc Length (formula from section 6.4)
s = ∫(lower
limit = a , upper limit = b) sqrt(1 + [f '(x)]^2)
dx
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Comparing 2 Fluid Forces ( General equation Section 6.7)
s = ∫(lower
limit = c , upper limit = d) h(y)*L(y) dy
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More links to help on Visual Calc HP:
Tutorial
on integration using the method of substitution.
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