Studied by 9 people

5.0(1)

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hint

1

∫0dx =

C

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2

∫[g(x) + h(x)]dx =

G(x) + H(x) + C

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3

∫xⁿdx =

1/n+1 * x^(n+1) + C

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4

∫e^xdx =

e^x + C

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5

∫sin(x)dx =

-cos(x) + C

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6

∫sec²(x)dx =

tan(x) + C

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7

∫sec(x)tan(x)dx =

sec(x) + C

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8

∫⅟√(1-x²)dx =

arcsin(x) + C

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9

∫kdx =

kx + C

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10

∫kf(x)dx =

kF(x) + C

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11

∫1/x dx =

ln(abs(x)) + C

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12

∫a^x dx =

a^x / lna + C

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13

∫cos(x)dx =

sin(x) + C

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14

∫csc²(x)dx =

-cot(x) + C

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15

∫csc(x)cot(x)dx =

-csc(x) + C

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16

∫⅟(1+x²) dx =

arctan(x) + C

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17

∫tan(x)dx =

ln(abs(sec(x))) + C

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18

∫cot(x)dx =

ln(abs(sin(x))) + C

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19

∫sec(x)dx =

ln(abs(secx + tanx)) + C

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20

∫csc(x)dx =

ln(abs(cscx - cotx)) + C

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21

∫ln(x)dx =

xlnx - x + C

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22

substitution for √a²-x²

x = a*sin(theta)

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23

substitution for √a²+x²

x = a*tan(theta)

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24

substitution for √x²-a²

x = a*sec(theta)

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25

1-sin²x =

cos^2 (x)

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26

1 + tan²x =

sec^2 (x)

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27

sec²x - 1 =

tan^2 (x)

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28

√a²+x² becomes

a*sec(theta)

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29

√x²-a² becomes

a*tan(theta)

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30

√a²-x² becomes

a*cos(theta)

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31

Integration by parts

integral(udv) = u*v - integral(v*du)

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32

LIATE

Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential

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33

sin(2x) =

2sin(x)cos(x)

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34

sin²x (power reduction) =

(1 - cos(2x))/2

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35

cos²x (power reduction) =

(1 + cos(2x))/2

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36

sinAcosB =

1/2(sin(A+B) + sin(A-B))

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37

sinAsinB =

1/2(cos(A-B) - cos(A+B))

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38

cosAcosB = `

1/2(cos(A-B) + cos(A+B))

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39

∫⅟(a²+x²) dx =

1/a * arctan(x/a) + C

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40

average value of f on [a, b]

1/(b-a) * integral from a to b of [f(x)dx]

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41

Arc length L of f(x) on [a, b]

L = ∫[sqrt(1 + (dy/dx)^2) dx] on [a,b]

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42

Surface area in terms of x across the x-axis

SA = ∫[2πf(x) * sqrt(1 + (dy/dx)^2) dx] on [a,b]

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43

Surface area in terms of x across the y-axis

SA = integral from a to b of [2 pi x * sqrt(1 + (dy/dx)^2) dx]

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44

Surface area in terms of y across the x-axis

SA = integral from a to b of [2 pi y * sqrt(1 + (dx/dy)^2) dy]

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45

Surface area in terms of y across the y-axis

SA = integral from a to b of [2 pi g(y) * sqrt(1 + (dx/dy)^2) dy]

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