Math 20B Integral Calculus
Lecture 12
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Tricks for trigs
Slide 1
• Review: Recursion formulas.
• Integral of some trigonometric functions.
• integrals of f : IR → C.
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Reduction formulas are a simple way to write
complicated integrals
In the case of the function sin(x) one has:
Slide 2
Z
1
(n − 1)
(sin(x))n dx = − (sin(x))(n−1) cos(x) +
n
n
Z
(cos(x))n dx =
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1
(n − 1)
(cos(x))(n−1) sin(x) +
n
n
Z
Z
(sin(x))(n−2) dx.
(cos(x))(n−2) dx.
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Math 20B Integral Calculus
Lecture 12
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Tricks for sines and cosines
Slide 3
Consider integrals of the form
Z
[sin(x)]m [cos(x)]n dx.
If m or n is odd, then the integral can be done by
substitution, recalling:
sin0 (x) = cos(x),
cos0 (x) = − sin(x),
sin2 (x) + cos2 (x) = 1.
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Tricks for sines and cosines
Consider integrals of the form
Z
[sin(x)]m [cos(x)]n dx.
Slide 4
If both m and n are even, then integral above can be
done recalling
1
sin2 (x) = [1 − cos(2x)],
2
1
cos2 (x) = [1 + cos(2x)].
2
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Math 20B Integral Calculus
Lecture 12
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Tricks for tangents and secants
Slide 5
Consider integrals of the form
Z
[tan(x)]n [sec(x)]m dx.
If n is odd or if m is even, then substitution recalling
tan0 (x) = sec2 (x),
sec0 (x) = tan(x) sec(x),
sec2 (x) − tan2 (x) = 1.
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More tricks on sines and cosines
The following integrals can be done with the associated
formula:
Slide 6
Z
Z
Z
sin(mx) cos(nx) dx
→
sin(mx) sin(nx) dx
→
cos(mx) cos(nx) dx
→
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1
[sin(α − β) + sin(α + β)];
2
1
sin(α) sin(β) = [cos(α − β) − cos(α + β)];
2
1
cos(α) cos(β) = [cos(α − β) + cos(α + β)].
2
sin(α) cos(β) =
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Math 20B Integral Calculus
Lecture 12
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Integration can be generalized to functions
f : IR → C
These functions have the form f (x) = f1 (x) + i f2 (x),
with f1 (x) and f2 (x) real functions.
Slide 7
Examples:
f (x) = 2x + i ln(x),
1
f 0 (x) = 2 + i ,
x
g(x) = tan(3x) + i e2x
g 0 (x) = 3 sec2 (3x) + 2i e2x .
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Riemann sums, integration, and the FTC can be
generalized to function f : IR → C
Z
Slide 8
Example:
f (x)dx =
Z
Z
f1 (x)dx + i
Z
f2 (x)dx.
1 iax
i
e = − eiax .
ia
a
Z
1
[x2 + i cos(x)]dx = x3 + i sin(x).
3
&
eiax dx =
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Math 20B Integral Calculus
Lecture 12
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Complex tricks for sine and cosine
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Euler formulas
eix = cos(x) + i sin(x),
e−ix = cos(x) − i sin(x),
Slide 9
imply that
1 ix
[e + e−ix ],
2
1 ix
sin(x) =
[e − e−ix ].
2i
cos(x) =
Therefore, integrals in sines and cosines can be
transformed into complex integrals for exponentials.
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