integration + differentiation

∫ ke^kx dx

∫ ke^kx dx = ke^kx + c

∫ - ke^kx dx

∫ - ke^kx dx = - ke^kx + c

∫ x^-1 dx

∫ x^-1 dx = ln|x| + c

∫ cosx dx

∫ cosx dx = sinx + c

∫ sinx dx

∫ sinx dx = - cosx + c

∫ - cosx dx

∫ - cosx dx = - sinx + c

∫ - sinx dx

∫ - sinx dx = cosx + c

∫ sec²x dx

∫ sec²x dx = tanx + c

∫ - sec²x dx

∫ - sec²x dx = - tanx + c

∫ cosec²x dx

∫ cosec²x dx = - cotx + c

∫ - cosec²x dx

∫ cosec²x dx = cotx + c

∫ cosecxcotx dx

∫ cosecxcotx dx = - cosecx + c

∫ - cosecxcotx dx

∫ - cosecxcotx dx = cosecx + c

∫ secxtanx dx

∫ secxtanx dx = secx + c

∫ - secxtanx dx

∫ - secxtanx dx = - secx + c

trapezium rule;

∫^b_a y dx ≈ ?

∫^b_a y dx ≈ ½ h (y₀ + 2 (y₁ + y₂ + … + y_n-1) + y_n)

• h = (b - a) / n

• n = number of columns (number of strips -1)

• y_i = f (a + ih)

what is h in the trapezium rule?

h = (b - a) / n

• ∫^b_a y dx

• n = number of columns (number of strips -1)

what is n in the trapezium rule?

n = number of columns (number of strips -1)

what values can we not integrate?

• sin²x

• cos²x

• cot²x

• sinxcosx

how do we integrate values we can’t integrate?

rewrite them using trig identities

how do you integrate sin²x ?

here

∫ sin²x dx

∫ sin²x dx = ½ x - ¼ x cosx + c

how do you integrate cos²x ?

here

∫ cos²x dx

here

how do you integrate cot²x ?

here

∫ cot²x dx

here

how do you integrate sinxcosx ?

here

∫ sinxcosx dx

here

INTEGRATION BY PARTS

∫ ab dx

∫ uv’ = uv - ∫ vu’

PRODUCT RULE

y = uv

dy / dx = vu’ + uv’

QUOTIENT RULE

y = u/v

dy / dx = (vu’ - uv’) / v²

∫ tanx dx

∫ tanx dx = ln|secx| + c

∫ - tanx dx

∫ - tanx dx = - ln|secx| + c

∫ secx dx

∫ secx dx = ln|secx + tanx| + c

∫ - secx dx

∫ - secx dx = - ln|secx + tanx| + c

∫ cotx dx

∫ cotx dx = ln|sinx| + c

∫ - cotx dx

∫ - cotx dx = - ln|sinx| + c

∫ cosecx dx

∫ cosecx dx = - ln|cosecx + cotx| + c

∫ - cosecx dx

∫ - cosecx dx = ln|cosecx + cotx| + c

how do you make a trapezium rule estimate more accurate?

• using more strips

• using more trapezia

what does x^-1 differentiate to?

- x^-2

what does ln x differentiate to?

x^-1

which is the tangent?

here

which is the normal?

here

what is the gradient of the tangent?

same as the curve

what is the gradient of the normal?

negative reciprocal of the curve

parts of a circle here

how do you estimate the gradient at a particular point?

• find coordinates at that point by putting into equation (if you don’t have them already)

• draw a tangent at that point

• pick two points - one being the point you draw the tangent at (x₂ , y₂), and the other being some point on the tangent (x₁ , y₁)

• (y₂ - y₁) / (x₂ - x₁)

what is NOT estimating the gradient?

by using the first derivative

how do you hypothesis about the gradient of a curve when x = p?

• find first derivative for gradient function

• substitute x for p

what do you use (y₂ - y₁) / (x₂ - x₁) for?

finding the gradient of a point using points on the tangent drawn from the point

what do you use √ (y₂ - y₁)² + (x₂ - x₁)² for?

what do you say when asked to comment about the relationship between gradients?

• as the points move closer to the particular point, A, where the gradient between two points are taken from, they tend towards the gradient at A

here

how can you deduce the gradient of a tangent at a particular point without having a graph?

here 12 a 3 b

12 a 4 b

what does h / δx represent in differentiation?

a small change in the value of x

how do we denote a small change in the value of x?

h / δx

how do you differentiate ax (x + b) ?

NOT using the chain rule. expand to ax² + abx and then go from there lolz

how do you differentiate this 12c 2 j

how do you differentiate this 12c 4

12 d 7 c

12 e 7 b

12f challenge

when is the function increasing on the interval [a, b]?

f’(x) ≥ 0, when a < x < b

when is the function decreasing on the interval [a, b]?

f’(x) ≤ 0, when a < x < b

how do you notate strictly increasing / decreasing

> / <

(without the equals to)

how do you find the values of x where a function is increasing?

1. differentiate term for first derivation

2. f’(x) ≥ 0

3. solve for x

4. ‘f(x) is decreasing when x ≥ a’

how do you find the values of x where a function is decreasing?

1. differentiate term for first derivation

2. f’(x) ≤ 0

3. solve for x

4. ‘f(x) is decreasing when x ≤ a’

how do you do u-sub?

1. always start by differentiating the substitution, u = bla bla bla

2. see what you can replace / cancel out by rearranging your du / dx function. you must rearrange for dx anywayz to replace the dx in the original integral function

3. rewrite function in terms of u, leaving some unsubstituted if they be cancelled out by another function

4. expand brackets for integrate-able function

5. integrate with respect to u (bcuz it’s du now)

indefinite:

1. rmbr the + c ! (if indefinite)

2. now sub your substitution back in !

definite:

1. rewrite the limits by putting them into the substitution function, giving you ‘u’ limits

2. do not substitute the function back in. instead, substitute your u limits into the integrated function and voila

how do you rewrite the limits of an indefinite integral in u-sub?

by putting them in the substitution function

in u-sub, what does u² differentiate to?

2u du / dx

in u-sub, what does √x differentiate to?

1 / (2 √x )

should you ever change the limit placement, even if your new limits (in u-sub) are not placed by size?

NO ! never change it.

what can you do when your denominator is a completing the square?

split it into two separate fractions with different denominators of each factor

how do you integrate the function x^-1?

1 / differential of x function x ln x