6.2 Trigonometric Symbols and Substitution Flashcards
1
Q
sin²(x) (even powers)
A
(1/2)(1-cos(2x))
2
Q
cos²(x) (even powers)
A
(1/2)(1+cos(2x))
3
Q
sin²(x) (odd powers)
A
(1 - cos²(x))
4
Q
cos²(x) (odd powers)
A
(1 - sin²(x))
5
Q
tan²(x) (even or odd powers)
A
sec²(x) - 1
6
Q
sec²(x) (even or odd powers)
A
tan²(x) + 1
7
Q
√a² + x²
A
x = a*tan(θ)
8
Q
√a² - x²
A
x = a*sin(θ)
9
Q
√x² - a²
A
x = a*sec(θ)
10
Q
√a² - a²sin²(θ)
A
a*cos(θ)
11
Q
√a² - a²tan²(θ)
A
a*sec(θ)
12
Q
√a² - a²sec²(θ)
A
a*tan(θ)
13
Q
range of a*sin(θ)
A
-(π/2)
14
Q
range of a*tan(θ)
A
-(π/2)
15
Q
range of a*sec(θ)
A
0
16
Q
Integrate even powers of sin or cos
A
- Use the double-angle identity
- Pull out constants
- Integrate (use u-sub if necessary)
17
Q
Integrate odd powers of sin or cos
A
- Split into even and odd power multiplied
- Use the additive identity on even power
- Integrate w/u-sub
18
Q
Integrate even powers of tan
A
- Use additive Identity
2. Integrate wlu-sub
19
Q
Integrate odd powers of tan
A
- Split into even and odd power multiplied
- Use the additive identity on even power
- Integrate w/u-sub
20
Q
Using trigonometric substitution
A
- solve for x w/respect to θ
- solve for dθ
- substitute in for x and dx
- simplify expressions using identities
- integrate w/respect to θ
- create reference triangle and solve for trig functions w/respect to x if needed
- re-sub x in for θ
