calc 2 (21b) trig identities, integrals, & relevant formulas

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66 Terms

1
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∫ xⁿ dx
xⁿ⁺¹/(n+1) + C (n ≠ −1)
2
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∫ sin(kx) dx
−(1/k) cos(kx) + C
3
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∫ cos(kx) dx
(1/k) sin(kx) + C
4
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∫ tan(kx) dx
(1/k) ln|sec(kx)| + C
5
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∫ xⁿ dx
xⁿ⁺¹/(n+1) + C, n ≠ −1
6
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∫ (1/x) dx
ln|x| + C
7
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∫ sin(kx) dx
−(1/k) cos(kx) + C
8
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∫ cos(kx) dx
(1/k) sin(kx) + C
9
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∫ tan(kx) dx
(1/k) ln|sec(kx)| + C
10
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∫ sec(kx) dx
(1/k) ln|sec(kx) + tan(kx)| + C
11
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∫ sec(kx)tan(kx) dx
(1/k) sec(kx) + C
12
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∫ sec²(kx) dx
(1/k) tan(kx) + C
13
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∫ e^{kx} dx
(1/k) e^{kx} + C
14
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∫ 1/(a² + x²) dx
(1/a) tan⁻¹(x/a) + C
15
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∫ 1/√(a² − x²) dx
sin⁻¹(x/a) + C
16
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∫ 1/(x√(x² − a²)) dx
(1/a) sec⁻¹(|x/a|) + C
17
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18
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Σ k
n(n+1)/2
19
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Σ k²
n(n+1)(2n+1)/6
20
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Σ k³
[n(n+1)/2]²
21
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22
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Average value of f
(1/(b−a)) ∫ₐᵇ f(x) dx
23
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FTC Part I
d/dx ∫ₐˣ f(t) dt = f(x)
24
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FTC Part II
∫ₐᵇ f(x) dx = F(b) − F(a)
25
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u-substitution
∫ f(g(x))g′(x) dx = ∫ f(u) du
26
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Even function integral
2∫₀ᵃ f(x) dx
27
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Odd function integral
0
28
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29
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Area between curves (x)
∫ (top − bottom) dx
30
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Area between curves (y)
∫ (right − left) dy
31
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32
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Disk method
π ∫ R(x)² dx
33
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Washer method
π ∫ (R² − r²) dx
34
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Shell method
2π ∫ (radius)(height) dx
35
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36
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Arc length (x)
∫ √(1 + (dy/dx)²) dx
37
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Arc length (y)
∫ √(1 + (dx/dy)²) dy
38
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39
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Surface area about x-axis
2π ∫ f(x) √(1 + (f′(x))²) dx
40
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41
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Definition of ln(x)
∫₁ˣ 1/t dt
42
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Derivative of aˣ
aˣ ln(a)
43
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44
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Integration by parts
∫ u dv = uv − ∫ v du
45
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46
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sin²x
(1 − cos(2x))/2
47
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cos²x
(1 + cos(2x))/2
48
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sin(2x)
2 sin x cos x
49
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cos(2x)
cos²x − sin²x
50
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1 + tan²x
sec²x
51
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1 + cot²x
csc²x
52
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53
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Trig sub: √(a² − x²)
x = a sinθ
54
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Trig sub: √(a² + x²)
x = a tanθ
55
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Trig sub: √(x² − a²)
x = a secθ
56
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57
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Trapezoidal Rule
(Δx/2)[f(x₀)+2Σf(xᵢ)+f(xₙ)]
58
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Simpson’s Rule
(Δx/3)[f(x₀)+4Σ f(odd)+2Σ f(even)+f(xₙ)]
59
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60
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Limit Comparison Test
lim (f(x)/g(x)) = L > 0 → same behavior
61
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62
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dy/dx for parametric
(dy/dt)/(dx/dt)
63
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Arc length (parametric)
∫ √[(dx/dt)² + (dy/dt)²] dt
64
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Area (parametric)
∫ y(t)x′(t) dt
65
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66
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Polar to Cartesian

x = r cosθ, y = r sinθ, tanθ=y/x