11. Integration Flashcards

1
Q

∫a(bx+d)^e

A

a/b X 1/(e+1) (bx+d)^e+1 + c

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2
Q

∫a/(bx+d)

A

a/b ln(|bx+d|) + c

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3
Q

∫a cos(bx+d)

A

a/b sin(bx+d) + c

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4
Q

∫a sin(bx+d)

A

-a/b cos(bx+d) + c

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5
Q

∫sec^2 x

A

tan(x) + c

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6
Q

∫cosec(x)cot(x)

A

-cosec(x) + c

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7
Q

∫cosec^2(x)

A

-cot(x) + c

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8
Q

∫sec(x)tan(x)

A

sec(x) + c

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9
Q

∫a((f(x))^n where a is a multiple of f’(x)

A

1/(n+1) X a/f’(x) X (f(x))^n+1 + c

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10
Q

∫af(x)/g(x) where f(x) is a multiple of g’(x)

A

af(x)/g’(x) ln(g(x))+ c

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11
Q

∫f(x) X e^g(x) where f(x) is a multiple of g’(x)

A

f(x)/g’(x) e^g(x) + c

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12
Q

Substitution for sin^2(x)

A

1/2 - 1/2(cos2x)

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13
Q

Substitution for cos^2(x)

A

1/2 + 1/2(cos2x)

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14
Q

Substitution for tan^2(x)

A

sec^2(x) - 1

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15
Q

Powers of sec and cosec

A

Don’t change in integration

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16
Q

Reverse chain rule

A

Where one component is a multiple of the single derivative of the second component
Add one to the power of the second component and differentiate
Put a multiple at the front so it goes to what you were integrating

17
Q

Integration by Substitution

A
  1. Rearrange for x in terms of u and dx in terms of du by finding du/dx
  2. Substitute in for x and dx and simplify
  3. Take the integral of that
  4. Write answer in terms of x (don’t need to expand)
18
Q

Definite Integration by Substitution

A

Find u for each x and integrate between those

19
Q

∫u(dv/dx) dx

A

uv - ∫v(du/dx) dx

20
Q

Integration by parts process

A
  1. Choose a value for u and dv/dx based on which you want to integrate and differentiate, leading to integrating the product of their results
  2. Find v and du/dx
  3. Substitute into the formula
21
Q

Repeated integration by parts

A

When you get a minus the original integral add that to both sides and half

22
Q

Integrating ln(x)

A

Integrate 1 ln(x) by parts

23
Q

When to use partial fractions

A

If the numerator is not related to the derivative of the denominator and the denominator can be written as the product of linear factors (check for quotient)

24
Q

Area between two curves

A

Subtract the lower equation from the higher or integrate separately and subtract depending on if it is possible

25
Setting up differential equations
set dy/dx or other proportional to what the question says replace with dy/dx = k... if it is decreasing use -k, k>0
26
Solving differential equations
1 Use the proportion to write the derivative in terms of k 2. Put all y terms on the LHS and x on the RHS with the respective d(variable) 3. Integrate and add a +c to one side 4. Rearrange into a nice/requested form 5. Substitute for c if required
27
Showing what happens as t increases, or showing a maximum value for differential equations
Show what happens as t tends to infinity
28
Trapezium rule
y = h/2 (y0 + 2(y1 + … + yn-1) + yn) | Where h is the x-length of each trapezium and y0 -> yn the y-coordinates for x0 -> xn
29
Finding h for trapezium rule
(b-a)/n where b and a are the maximum and minimum limits and n the number of trapeziums
30
Parametric integration
Multiply the y equation by the differential of the x one and integrate with respect to t Integrate between limits of t
31
Integration as the limit of the sum
Summing a function of x δx (lim δ->0) is the same as integrating that function between the limits