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The MF19 Formula Sheet

The exact formulae Cambridge prints for you in the 9231 exam — and, just as important, the ones it doesn't, that you must commit to memory. Search to find anything fast.

GIVEN On the MF19 sheet — Further Pure

Provided in the exam. Know how to use them; you don't need to memorise them.

Summations

Sum of r

\sum_{r = 1}^{n} r = \frac{1}{2} n (n + 1)

Sum of r²

\sum_{r = 1}^{n} r^{2} = \frac{1}{6} n (n + 1) (2 n + 1)

Sum of r³

\sum_{r = 1}^{n} r^{3} = \frac{1}{4} n^{2} (n + 1)^{2}

Maclaurin's series

General form

f (x) = f (0) + x f^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + \dots + \frac{x^{r}}{r!} f^{(r)} (0) + \dots

eˣ

e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{r}}{r!} + \dots

(all x)

ln(1+x)

\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots + (- 1)^{r + 1} \frac{x^{r}}{r} + \dots

(−1<x≤1)

sin x

\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots

(all x)

cos x

\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots

(all x)

tan⁻¹x

\tan^{- 1} x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots

(−1≤x≤1)

sinh x

\sinh x = x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \dots

(all x)

cosh x

\cosh x = 1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \dots

(all x)

tanh⁻¹x

\tanh^{- 1} x = x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \dots

(−1<x<1)

Trigonometry — the t = tan(½x) substitution

sin x

\sin x = \frac{2 t}{1 + t^{2}}

cos x

\cos x = \frac{1 - t^{2}}{1 + t^{2}}

Hyperbolic functions

Identity

\cosh^{2} x - \sinh^{2} x = 1

Double angle

\sinh 2 x = 2 \sinh x \cosh x, \cosh 2 x = \cosh^{2} x + \sinh^{2} x

sinh⁻¹x

\sinh^{- 1} x = \ln (x + \sqrt{x^{2} + 1})

cosh⁻¹x

\cosh^{- 1} x = \ln (x + \sqrt{x^{2} - 1})

(x≥1)

tanh⁻¹x

\tanh^{- 1} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}

(|x|<1)

Standard derivatives

sin⁻¹x

\frac{1}{\sqrt{1 - x^{2}}}

cos⁻¹x

- \frac{1}{\sqrt{1 - x^{2}}}

tan⁻¹x

\frac{1}{1 + x^{2}}

sinh x

\cosh x

cosh x

\sinh x

tanh x

{sech}^{2} x

sinh⁻¹x

\frac{1}{\sqrt{1 + x^{2}}}

cosh⁻¹x

\frac{1}{\sqrt{x^{2} - 1}}

tanh⁻¹x

\frac{1}{1 - x^{2}}

Standard integrals (constants omitted; a > 0)

∫ sec x

\ln | \sec x + \tan x |

∫ cosec x

- \ln | cosec x + \cot x |

∫ sinh x

\cosh x

∫ cosh x

\sinh x

∫ sech²x

\tanh x

∫ 1/√(a²−x²)

\sin^{- 1} \frac{x}{a}

(|x|<a)

∫ 1/√(x²−a²)

\cosh^{- 1} \frac{x}{a}

(x>a)

∫ 1/√(x²+a²)

\sinh^{- 1} \frac{x}{a}

Also on the sheet — from Pure Maths (9709, assumed)

∫ 1/(a²+x²)

\frac{1}{a} \tan^{- 1} \frac{x}{a}

∫ 1/(a²−x²)

\frac{1}{2 a} \ln | \frac{a + x}{a - x} |

(|x|<a)

∫ 1/(x²−a²)

\frac{1}{2 a} \ln | \frac{x - a}{x + a} |

(|x|>a)

By parts

\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x

MEMORISE NOT on the sheet — you must know these

Cambridge does not give these. Losing them costs whole questions. Drill until automatic.

de Moivre

(\cos θ + i \sin θ)^{n} = \cos n θ + i \sin n θ

n th roots

w_{k} = r^{1 / n} cis \frac{θ + 2 π k}{n}, k = 0, \dots, n - 1

Roots ↔ coeffs (quadratic)

α + β = - \frac{b}{a}, α β = \frac{c}{a}

Roots ↔ coeffs (cubic)

\sum α = - \frac{b}{a}, \sum α β = \frac{c}{a}, α β γ = - \frac{d}{a}

Polar area

A = \frac{1}{2} \int r^{2} d θ

Vector product

| 𝐚 \times 𝐛 | = | 𝐚 | | 𝐛 | \sin θ

2×2 inverse

{(\begin{array}{cc} a & b \\ c & d \end{array})}^{- 1} = \frac{1}{a d - b c} (\begin{array}{cc} d & - b \\ - c & a \end{array})

Rotation about O, angle θ

(\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array})

Reflection in line at θ to x-axis

(\begin{array}{cc} \cos 2 θ & \sin 2 θ \\ \sin 2 θ & - \cos 2 θ \end{array})

Area scale factor

area SF = | \det 𝐌 |

Invariant lines y = mx

b m^{2} + (a - d) m - c = 0

Reversal law

(𝐀 𝐁)^{- 1} = 𝐁^{- 1} 𝐀^{- 1}

Eigenvalues

\det (𝐀 - λ 𝐈) = 0

Roots of unity

1 + ω + ω^{2} + \dots + ω^{n - 1} = 0

Integrating factor

IF = e^{\int P d x}

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