Integration/differentiation

Question 1

What is the integral of Sec^2x

Answer 1

tanx

Question 2

What is the integral of cosecxcotx

Answer 2

-cosecx

Question 3

What is the integral of cosec^2x

Answer 3

-cotx

Question 4

What is the integral of secxtanx

Answer 4

secx

Question 5

Question 6

How do we integrate this parametrically

Answer 6

Question 7

How do we differentiate 2xy with respect to x

Answer 7

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

Question 8

What is the formula for parametric integration

Answer 8

Question 9

What’s the formula for integration by parts

Answer 9

Question 10

Prove the derivative of cosx is -sinx

Question 11

What is the first principles equation

Answer 11

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

Question 12

Prove the derivative of sinx is cosx

Question 14

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

Answer 14

We put a minus in front of the function and flip

Question 15

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

Answer 15

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)

Question 16

What is the first principles equation

Answer 16

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

Question 17

What’s the derivative of

Answer 17

Question 18

What’s the derivative of

Answer 18

Question 19

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

Answer 19

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

Question 20

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

Answer 20

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

Question 21

Question 22

How do we confirm a stationary point

Answer 22

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

Question 23

How to get from this to that

Answer 23

Question 24

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

Answer 24

Separate the integral like this

Question 25

Answer 25

Only look at first line for each section

Question 26

When integrating OR DIFFERENTIATING sin²x/cos²x, what must we do

Answer 26

Use the cos2x identity to rewrite it then integrate or differentiate

Question 27

When using trapezium rule when is it an overestimate and when is it an underestimate

Answer 27

Concave---> underestimate because top of trapezia lies below the curve Convex ----> Overestimate because top of trapezia lie above the curve