Formulas to memorise for CT4

Question 1

Antiderivative notation

Answer 1

∫ f(x) dx

Question 2

Antiderivative formula

Answer 2

∫ axⁿ dx = [axⁿ⁺¹ / n + 1] + c

where n ≠ -1

Question 3

Integration chain rule

Answer 3

∫ k(ax + b)ⁿ dx

{ [ k(ax + b)ⁿ⁺¹ ] / a(n + 1) } + c

Question 4

Definite integral formula

Answer 4

∫ ᵇₐ f’ (x) dx = [f(x)ᵇₐ] = f(b) - f(a) where b > a

Question 5

Area bounded by a curve and line or by two curves

Answer 5

A = ∫ ᵇₐ [f(x) - g(x)] dx where b > a

Question 6

Area enclosed by a curve and the x-axis

Answer 6

∫ᵇₐ f(x) dx where b > a

Question 7

Area enclosed by a curve and the y-axis

Answer 7

∫ᵇₐ f(y) dy where b > a

Question 8

Area above and below the x-axis

Answer 8

∫ᵇₐ f(x) dx + | ∫ᶜᵦ f(x) dx | where c > b > a

Question 9

Volumes of solids around x-axis

Answer 9

π ∫ᵇₐ f(x)² dx

where b > a and f(x)² is expanded before differentiating

Question 10

Volumes of solids around y-axis

Answer 10

π ∫ᵇₐ f(y)² dy

where b > a and f(y)² is expanded before differentiating

Question 11

Binomial Distribution; notation, formula, expected value, variance and mode

Answer 11

X ~ B (n, p)

P(X = x) = (n x*)pˣqⁿ⁻ˣ

• x placed below n
• same as nCr on calculator

E(X) = np | | Var(X) = npq

Mode is trial with > E(X)

Question 12

Geometric Distribution; notation, formula, expected value, variance and mode

Answer 12

X ~ Geo (p)

P(X = x) = pqˣ⁻¹

E(X) = 1/p | | Var(X) = q/p²

Mode is always trial 1

Question 13

Geometric Distributional Inequalities; fewer than formula,

after formula

Answer 13

P(X ≤ x)
= 1 - qˣ

P(X > x)
= qˣ

Question 14

Binomial expansion; term bʳ, term aʳ

Answer 14

(nCr)aⁿ⁻ʳbʳ

[nC(n-r)]aʳbⁿ⁻ʳ

Question 15

Binomial theorem

formula
term bʳ
term aʳ

Answer 15

(nC0)aⁿb⁰ + (nC1)aⁿ⁻¹b¹ + (nC2)aⁿ⁻²b² + … + (nCn)a⁰bⁿ

(nCr)aⁿ⁻ʳbʳ

[nC(n-r)]aʳbⁿ⁻ʳ