Math 1B
§ 7.2 Trigonometric Integrals
Overview: In this section we use trigonometric identities to integrate certain combinations of
trigonometric functions. Here are some useful trig identities:
→
sin 2 x + cos 2 x = 1
and
sin 2 x = 1− cos 2 x
cos2 x = 1− sin 2 x
tan 2 x + 1 = sec 2 x
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1+ cot 2 x = csc 2 x
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cos2 x =
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1
(1+ cos2x )
2
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sin 2 x =
1
(1− cos2x )
2
sin2x = 2sin x cos x
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In addition to the Table of Integrals, there is another integral that comes up often enough to make it
worth remembering:
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∫ sec x dx = ln sec x + tan x + C
Here is where this comes from:
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Example: Find
∫ sin
3
x dx
∫ sin
4
x cos 5 x dx
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Example: Find
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2
π
2
Example: Evaluate
∫ cos
2
x dx
0
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Example: Find
∫ sin
2
x cos 2 x dx
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Strategy:
∫ sin
m
x cos n x dx
1. If the power of cosine is odd, break off one factor of cos x and write the rest in terms of sine. Then
let u = sin x .
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2. If the power of sine is odd, break off one factor of sin x and write the rest in terms of cosine. Then
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let u = cos x .
Note: If both powers are odd, use either plan.
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3. If the powers of both sine and cosine are even, use sin 2 x =
Example: Find
∫ tan
2
x sec 4 x dx
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1
1
(1− cos2x ) and cos2 x = (1+ cos2x ) .
2
2
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Example: Find
∫ tan
3
x sec 5 x dx
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Strategy:
∫ tan
m
x sec n x dx
1. If the power of secant is even, break off one factor of sec 2 x and write the rest in terms of tangent.
Then let u = tan x .
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2. If the power of tangent is odd, break off one factor of sec x tan x and write the rest in terms of
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secant. Then let u = sec x .
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Another useful integral:
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∫ sec
3
x dx =
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1
(sec x tan x + ln sec x + tan x ) + C
2
Read the example in the book (Pg. 475) to see where this comes from.
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More useful identities:
1
[sin( A − B) + sin( A + B)]
2
1
sin Asin B = [cos( A − B) − cos( A + B)]
2
1
cos Acos B = [cos( A − B) + cos( A + B)]
2
sin Acos B =
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Example: Find
∫ sin 7x cos 3x dx
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