Integration/differentiation Flashcards

(27 cards)

1
Q

What is the integral of Sec^2x

A

tanx

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2
Q

What is the integral of cosecxcotx

A

-cosecx

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3
Q

What is the integral of cosec^2x

A

-cotx

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4
Q

What is the integral of secxtanx

A

secx

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5
Q
A
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6
Q

How do we integrate this parametrically

A
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7
Q

How do we differentiate 2xy with respect to x

A

Split x’s & ys and apply product rule
ie
2x x y

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8
Q

What is the formula for parametric integration

A
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9
Q

What’s the formula for integration by parts

A
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10
Q

Prove the derivative of cosx is -sinx

A
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11
Q

What is the first principles equation

A

f(x+h) - f(x)/h

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12
Q

Prove the derivative of sinx is cosx

A
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13
Q
A
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14
Q

What do we do when integrating if we want to flip the bounds (if top is smaller than bottom)

A

We put a minus in front of the function and flip

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15
Q

What’s the pneumonic to remember order of priority of what is U and what is V’ when integrating by parts

A

IF ITS A LOGARITHM, ITS U, IF NO LOG THEN ITS THE ALGEBRAIC TERM (x term) FOR U

🧠 How to Choose
𝑢 and 𝑑𝑣

A useful rule is LIATE:

Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

(We don’t integrate inverse trig functions in a level maths so the rule to remember is LOG TAKES PRIORITY FOR U)

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16
Q

What is the first principles equation

A

f(x+h) - f(x)/h

17
Q

What’s the derivative of

18
Q

What’s the derivative of

19
Q

If second derivative is 0, what do we do to confirm the nature of the stationary point

A

If second derivative is negative –> Local MAX
If second derivative is positive —> Local MIN

If second derivative is 0 *probably POI:
Check SECOND DERIVATIVE either side of x value:

Changes sign either side (from + to – or – to +) then point of inflection

Negative and negative –> May be a Local MAX but have to check first derivative: Positive to negative means MAX
Positive to positive —> May be a Local MIN but have to check first derivative: Negative to positive means MIN

20
Q

What do we do when we have y = sinx…. and x=cos…..
(Or some other variation including sin and cos), when we want to covert parametric to cartesian

A

If its around this form always square the x and y to get sin²x and cos²x squared, then used sin²x + cos²x = 1 to connect the 2 equations somehow

22
Q

How do we confirm a stationary point

A

Check SECOND DERIVATIVE either side of x value
But also must show second derivative is 0 first

23
Q

How to get from this to that

24
Q

When we have a question like part be, what do we do

A

Separate the integral like this

25
Only look at first line for each section
26
When integrating OR DIFFERENTIATING sin²x/cos²x, what must we do
Use the cos2x identity to rewrite it then integrate or differentiate
27
When using trapezium rule when is it an overestimate and when is it an underestimate
Concave---> underestimate because top of trapezia lies below the curve Convex ----> Overestimate because top of trapezia lie above the curve