This is a draft cheat sheet. It is a work in progress and is not finished yet.
Trig Integrals
∫sinx dx |
= -cosx dx + C |
∫cosx dx |
= sinx dx + C |
∫sec2x dx |
= tanx dx + C |
∫tanx dx |
= ln|secx| + C |
∫secx tanx dx |
= secx + C |
∫csc2x dx |
= -cotx + C |
∫cscx cotx dx |
= -cscx + C |
∫cotx dx |
= ln|sinx| + C |
Trig Identities
∫(1/x2 + a2) dx |
= 1/a tan-1(x/a) + C |
∫(1/Sqrt(a2 - x2) dx |
= sin-1(x/a) + C |
Area Between Curves
Area = ∫[Height] Width
A = ∫(f(x) - g(x)] dx
1. Graph Equasions
2. Label
3. Determine how to slice
4. Set up dA
5. dA = height*dx
6. Get range a & b from intersections
7. Plug in and find area
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Volume by Disk
dV = A(x) dx
V = ∫A(x)dx
Volume = ∫Radius2 * Thickness
V = ∫(pi(r)2) dx
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Volume by Washer
dV = A(x) dx
V = ∫A(x) dx
Volume = ∫[(pi r out2)-(pi r in2)] dx
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Slice Perpendicular to Axis of Rotation
Volume by Shell
dVolume = Circumference * dArea
dV = (2 pi r) dArea
V = ∫(2 pi r)(Area)dx
1. Write: dV = 2 pi r dA
2. Find dA(height dx)
3. Find Radius(x or y)
4. Plug in
5. Take integral
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Slice Parallel to Axis of Rotation
Average Value of a Function
Average Value = 1/b-a * ∫f(x) dx
Symmetry:
If f(x) is EVEN, then ∫f(x)dx from -a to a = 2∫f(x) from 0 to a
If f(x) is ODD, then ∫f(x)dx from -a to a = 0
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Important Integrals
∫c f(x) dx |
= c ∫f(x) dx |
∫[f(x) + g(x)] dx |
= ∫f(x) dx + ∫g(x) dx |
∫ 1/x dx |
= ln|x| + C |
∫ex dx |
= ex + C |
∫bx dx |
= (bx / lnb) + C |
Methods of Integration
Method |
When to Use |
Example |
U-Substitution |
When a Polynomial is
raised to a power > 1 |
∫(3x + 5)5 |
Integration by Parts |
When U-Sub will not work |
∫xex |
Trigonometric Integration |
Only Trig raised to powers |
∫sin6x cos3xdx |
Trigonometric Substitution |
3/2 powers or Sqrt(a2-x2) etc. |
dx/(x2Sqrt(25-x2)) |
Integration by Parts
Logarithmic
Inverse trig
Algebraic
Trigonometric
Exponential
∫u dv = u v - ∫v du
1. Write u v - ∫v du
2. Use LIATE to find u; the other term becomes dV
3. Setup u= dV= du= V=
4. Solve
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Cyclical Functions will need to be split and substituted.
Trigonometric Integration
Identities
sin2t+cos2t = 1
sin2t = 1/2 [1-cos(2t)]
cos2t = 1/2 [1+cos(2t)]
Can use with U-Substitution
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Don't change all of the trig to the same form.
Trigonometric Integration
Identities
sin2t+cos2t = 1
sin2t = 1/2 [1-cos(2t)]
cos2t = 1/2 [1+cos(2t)]
Can use with U-Substitution
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Don't change all of the trig to the same form.
Trigonometric Substitution
Pythag. Identities
sin2 + cos2 = 1
1 + tan2 = sec2
1 + cot2 = csc2
1. Identify a and u
2. Sub in the trig
3. Manipulate to simplify
4. Get rid of trig with a triangle
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