IB Maths AA formula booklet, explained

Every formula from the prior learning page and the five topics, word for word and in the order the IB prints them. Press Explain beside an entry to see what it means and how to use it, with worked examples for the formulas.

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Booklet
Mathematics: analysis and approaches formula booklet, first examinations 2021, version 1.0
Syllabus
Mathematics: Analysis and Approaches, first assessment 2021
Levels
SL and HL, switch between them above the booklet
In the exam
You get the official printed copy, not this page and not our explanations
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Mathematics: analysis and approaches formula booklet

First examinations 2021, Version 1.0

Reproduced from the IB Mathematics: analysis and approaches formula booklet, First examinations 2021, Version 1.0. Explanations are ours.

Contents

  1. Prior learningsheet 3
  2. Topic 1: Number and algebrasheet 4
  3. Topic 2: Functionssheet 5
  4. Topic 3: Geometry and trigonometrysheet 6
  5. Topic 4: Statistics and probabilitysheet 7
  6. Topic 5: Calculussheet 8

Prior learning

SL and HL

Area of a parallelogram
A=bhA = bh
where bb is the base, hh is the height
Area of a triangle
A=12(bh)A = \frac{1}{2}(bh)
where bb is the base, hh is the height
Area of a trapezoid
A=12(a+b)hA = \frac{1}{2}(a + b)h
where aa and bb are the parallel sides, hh is the height
Area of a circle
A=πr2A = \pi r^2
where rr is the radius
Circumference of a circle
C=2πrC = 2\pi r
where rr is the radius
Volume of a cuboid
V=lwhV = lwh
where ll is the length, ww is the width, hh is the height
Volume of a cylinder
V=πr2hV = \pi r^2 h
where rr is the radius, hh is the height
Volume of a prism
V=AhV = Ah
where AA is the area of cross-section, hh is the height
Area of the curved surface of a cylinder
A=2πrhA = 2\pi rh
where rr is the radius, hh is the height
Distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2)
d=(x1−x2)2+(y1−y2)2d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
Coordinates of the midpoint of a line segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2)
(x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Topic 1: Number and algebra

SL and HL

SL 1.2
The nnth term of an arithmetic sequence
un=u1+(n−1)du_n = u_1 + (n - 1)d
SL 1.2
The sum of nn terms of an arithmetic sequence
Sn=n2(2u1+(n−1)d); Sn=n2(u1+un)S_n = \frac{n}{2}\left(2u_1 + (n - 1)d\right);\ S_n = \frac{n}{2}(u_1 + u_n)
SL 1.3
The nnth term of a geometric sequence
un=u1rn−1u_n = u_1 r^{n-1}
SL 1.3
The sum of nn terms of a finite geometric sequence
Sn=u1(rn−1)r−1=u1(1−rn)1−rS_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}
, r≠1r \neq 1
SL 1.4
Compound interest
FV=PV×(1+r100k)knFV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}
where FVFV is the future value, PVPV is the present value, nn is the number of years, kk is the number of compounding periods per year, r%r\% is the nominal annual rate of interest
SL 1.5
Exponents and logarithms
ax=b⇔x=log⁡aba^x = b \Leftrightarrow x = \log_a b
where a>0a > 0, b>0b > 0, a≠1a \neq 1
SL 1.7
Exponents and logarithms
log⁡axy=log⁡ax+log⁡ay\log_a xy = \log_a x + \log_a y
log⁡axy=log⁡ax−log⁡ay\log_a \frac{x}{y} = \log_a x - \log_a y
log⁡axm=mlog⁡ax\log_a x^m = m\log_a x
log⁡ax=log⁡bxlog⁡ba\log_a x = \frac{\log_b x}{\log_b a}
SL 1.8
The sum of an infinite geometric sequence
S∞=u11−rS_{\infty} = \frac{u_1}{1 - r}
, ∣r∣<1|r| < 1
SL 1.9
Binomial theorem
(a+b)n=an+nC1 an−1b+…+nCr an−rbr+…+bn(a + b)^n = a^n + {}^{n}\mathrm{C}_{1}\,a^{n-1}b + \ldots + {}^{n}\mathrm{C}_{r}\,a^{n-r}b^r + \ldots + b^n
nCr=n!r!(n−r)!{}^{n}\mathrm{C}_{r} = \frac{n!}{r!(n - r)!}

HL only

AHL 1.10
Combinations
nCr=n!r!(n−r)!{}^{n}\mathrm{C}_{r} = \frac{n!}{r!(n - r)!}
AHL 1.10
Permutations
nPr=n!(n−r)!{}^{n}\mathrm{P}_{r} = \frac{n!}{(n - r)!}
AHL 1.12
Complex numbers
z=a+biz = a + b\mathrm{i}
AHL 1.13
Modulus-argument (polar) and exponential (Euler) form
z=r(cos⁡θ+isin⁡θ)=reiθ=rcis⁡θz = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta
AHL 1.14
De Moivre’s theorem
[r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ)=rneinθ=rncis⁡nθ\left[r(\cos\theta + \mathrm{i}\sin\theta)\right]^n = r^n(\cos n\theta + \mathrm{i}\sin n\theta) = r^n\mathrm{e}^{\mathrm{i}n\theta} = r^n\operatorname{cis} n\theta

Topic 2: Functions

SL and HL

SL 2.1
Equations of a straight line
y=mx+c; ax+by+d=0; y−y1=m(x−x1)y = mx + c;\ ax + by + d = 0;\ y - y_1 = m(x - x_1)
SL 2.1
Gradient formula
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
SL 2.6
Axis of symmetry of the graph of a quadratic function
f(x)=ax2+bx+c⇒axis of symmetry is x=−b2af(x) = ax^2 + bx + c \Rightarrow \text{axis of symmetry is } x = -\frac{b}{2a}
SL 2.7
Solutions of a quadratic equation
ax2+bx+c=0⇒x=−b±b2−4ac2aax^2 + bx + c = 0 \quad\Rightarrow\quad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
, a≠0a \neq 0
SL 2.7
Discriminant
Δ=b2−4ac\Delta = b^2 - 4ac
SL 2.9
Exponential and logarithmic functions
ax=exln⁡a; log⁡aax=x=alog⁡axa^x = \mathrm{e}^{x\ln a};\ \log_a a^x = x = a^{\log_a x}
where a,x>0a, x > 0, a≠1a \neq 1

HL only

AHL 2.12
Sum and product of the roots of polynomial equations of the form ∑r=0narxr=0\sum_{r=0}^{n} a_r x^r = 0
Sum is −an−1an; product is (−1)na0an\text{Sum is } \frac{-a_{n-1}}{a_n};\ \text{product is } \frac{(-1)^n a_0}{a_n}

Topic 3: Geometry and trigonometry

SL and HL

SL 3.1
Distance between two points (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2)
d=(x1−x2)2+(y1−y2)2+(z1−z2)2d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}
SL 3.1
Coordinates of the midpoint of a line segment with endpoints (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2)
(x1+x22,y1+y22,z1+z22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)
SL 3.1
Volume of a right-pyramid
V=13AhV = \frac{1}{3}Ah
where AA is the area of the base, hh is the height
SL 3.1
Volume of a right cone
V=13πr2hV = \frac{1}{3}\pi r^2 h
where rr is the radius, hh is the height
SL 3.1
Area of the curved surface of a cone
A=πrlA = \pi rl
where rr is the radius, ll is the slant height
SL 3.1
Volume of a sphere
V=43πr3V = \frac{4}{3}\pi r^3
where rr is the radius
SL 3.1
Surface area of a sphere
A=4πr2A = 4\pi r^2
where rr is the radius
SL 3.2
Sine rule
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
SL 3.2
Cosine rule
c2=a2+b2−2abcos⁡C; cos⁡C=a2+b2−c22abc^2 = a^2 + b^2 - 2ab\cos C;\ \cos C = \frac{a^2 + b^2 - c^2}{2ab}
SL 3.2
Area of a triangle
A=12absin⁡CA = \frac{1}{2}ab\sin C
SL 3.4
Length of an arc
l=rθl = r\theta
where rr is the radius, θ\theta is the angle measured in radians
SL 3.4
Area of a sector
A=12r2θA = \frac{1}{2}r^2\theta
where rr is the radius, θ\theta is the angle measured in radians
SL 3.5
Identity for tan⁡θ\tan\theta
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
SL 3.6
Pythagorean identity
cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1
SL 3.6
Double angle identities
sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta
cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

HL only

AHL 3.9
Reciprocal trigonometric identities
sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}
cosec⁡θ=1sin⁡θ\operatorname{cosec}\theta = \frac{1}{\sin\theta}
AHL 3.9
Pythagorean identities
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
1+cot⁡2θ=cosec⁡2θ1 + \cot^2\theta = \operatorname{cosec}^2\theta
AHL 3.10
Compound angle identities
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
AHL 3.10
Double angle identity for tan
tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}
AHL 3.12
Magnitude of a vector
∣v∣=v12+v22+v32|\boldsymbol{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}
where v=(v1v2v3)\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}
AHL 3.13
Scalar product
v⋅w=v1w1+v2w2+v3w3\boldsymbol{v} \cdot \boldsymbol{w} = v_1w_1 + v_2w_2 + v_3w_3
where v=(v1v2v3)\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, w=(w1w2w3)\boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}
v⋅w=∣v∣∣w∣cos⁡θ\boldsymbol{v} \cdot \boldsymbol{w} = |\boldsymbol{v}||\boldsymbol{w}|\cos\theta
where θ\theta is the angle between v\boldsymbol{v} and w\boldsymbol{w}
AHL 3.13
Angle between two vectors
cos⁡θ=v1w1+v2w2+v3w3∣v∣∣w∣\cos\theta = \frac{v_1w_1 + v_2w_2 + v_3w_3}{|\boldsymbol{v}||\boldsymbol{w}|}
AHL 3.14
Vector equation of a line
r=a+λb\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b}
AHL 3.14
Parametric form of the equation of a line
x=x0+λl, y=y0+λm, z=z0+λnx = x_0 + \lambda l,\ y = y_0 + \lambda m,\ z = z_0 + \lambda n
AHL 3.14
Cartesian equations of a line
x−x0l=y−y0m=z−z0n\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}
AHL 3.16
Vector product
v×w=(v2w3−v3w2v3w1−v1w3v1w2−v2w1)\boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}
where v=(v1v2v3)\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, w=(w1w2w3)\boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}
∣v×w∣=∣v∣∣w∣sin⁡θ|\boldsymbol{v} \times \boldsymbol{w}| = |\boldsymbol{v}||\boldsymbol{w}|\sin\theta
where θ\theta is the angle between v\boldsymbol{v} and w\boldsymbol{w}
AHL 3.16
Area of a parallelogram
A=∣v×w∣A = |\boldsymbol{v} \times \boldsymbol{w}|
where v\boldsymbol{v} and w\boldsymbol{w} form two adjacent sides of a parallelogram
AHL 3.17
Vector equation of a plane
r=a+λb+μc\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b} + \mu\boldsymbol{c}
AHL 3.17
Equation of a plane (using the normal vector)
r⋅n=a⋅n\boldsymbol{r} \cdot \boldsymbol{n} = \boldsymbol{a} \cdot \boldsymbol{n}
AHL 3.17
Cartesian equation of a plane
ax+by+cz=dax + by + cz = d

Topic 4: Statistics and probability

SL and HL

SL 4.2
Interquartile range
IQR=Q3−Q1\mathrm{IQR} = Q_3 - Q_1
SL 4.3
Mean, xˉ\bar{x}, of a set of data
xˉ=∑i=1kfixin\bar{x} = \frac{\sum_{i=1}^{k} f_i x_i}{n}
where n=∑i=1kfin = \sum_{i=1}^{k} f_i
SL 4.5
Probability of an event AA
P(A)=n(A)n(U)\mathrm{P}(A) = \frac{n(A)}{n(U)}
SL 4.5
Complementary events
P(A)+P(A′)=1\mathrm{P}(A) + \mathrm{P}(A') = 1
SL 4.6
Combined events
P(A∪B)=P(A)+P(B)−P(A∩B)\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)
SL 4.6
Mutually exclusive events
P(A∪B)=P(A)+P(B)\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B)
SL 4.6
Conditional probability
P(A∣B)=P(A∩B)P(B)\mathrm{P}(A \mid B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}
SL 4.6
Independent events
P(A∩B)=P(A) P(B)\mathrm{P}(A \cap B) = \mathrm{P}(A)\,\mathrm{P}(B)
SL 4.7
Expected value of a discrete random variable XX
E(X)=∑x P(X=x)\mathrm{E}(X) = \sum x\,\mathrm{P}(X = x)
SL 4.8
Binomial distribution X∼B(n,p)X \sim \mathrm{B}(n, p)
SL 4.8
Mean
E(X)=np\mathrm{E}(X) = np
SL 4.8
Variance
Var(X)=np(1−p)\mathrm{Var}(X) = np(1 - p)
SL 4.12
Standardized normal variable
z=x−μσz = \frac{x - \mu}{\sigma}

HL only

AHL 4.13
Bayes’ theorem
P(B∣A)=P(B) P(A∣B)P(B) P(A∣B)+P(B′) P(A∣B′)\mathrm{P}(B \mid A) = \frac{\mathrm{P}(B)\,\mathrm{P}(A \mid B)}{\mathrm{P}(B)\,\mathrm{P}(A \mid B) + \mathrm{P}(B')\,\mathrm{P}(A \mid B')}
P(Bi∣A)=P(Bi)P(A∣Bi)P(B1)P(A∣B1)+P(B2)P(A∣B2)+P(B3)P(A∣B3)\mathrm{P}(B_i \mid A) = \frac{\mathrm{P}(B_i)\mathrm{P}(A \mid B_i)}{\mathrm{P}(B_1)\mathrm{P}(A \mid B_1) + \mathrm{P}(B_2)\mathrm{P}(A \mid B_2) + \mathrm{P}(B_3)\mathrm{P}(A \mid B_3)}
AHL 4.14
Variance σ2\sigma^2
σ2=∑i=1kfi(xi−μ)2n=∑i=1kfixi2n−μ2\sigma^2 = \frac{\sum_{i=1}^{k} f_i(x_i - \mu)^2}{n} = \frac{\sum_{i=1}^{k} f_i x_i^2}{n} - \mu^2
AHL 4.14
Standard deviation σ\sigma
σ=∑i=1kfi(xi−μ)2n\sigma = \sqrt{\frac{\sum_{i=1}^{k} f_i(x_i - \mu)^2}{n}}
AHL 4.14
Linear transformation of a single random variable
E(aX+b)=aE(X)+b\mathrm{E}(aX + b) = a\mathrm{E}(X) + b
Var(aX+b)=a2 Var(X)\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)
AHL 4.14
Expected value of a continuous random variable XX
E(X)=μ=∫−∞∞xf(x) dx\mathrm{E}(X) = \mu = \int_{-\infty}^{\infty} x f(x)\,\mathrm{d}x
AHL 4.14
Variance
Var(X)=E(X−μ)2=E(X2)−[E(X)]2\mathrm{Var}(X) = \mathrm{E}(X - \mu)^2 = \mathrm{E}(X^2) - \left[\mathrm{E}(X)\right]^2
AHL 4.14
Variance of a discrete random variable XX
Var(X)=∑(x−μ)2 P(X=x)=∑x2 P(X=x)−μ2\mathrm{Var}(X) = \sum (x - \mu)^2\,\mathrm{P}(X = x) = \sum x^2\,\mathrm{P}(X = x) - \mu^2
AHL 4.14
Variance of a continuous random variable XX
Var(X)=∫−∞∞(x−μ)2f(x) dx=∫−∞∞x2f(x) dx−μ2\mathrm{Var}(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x)\,\mathrm{d}x = \int_{-\infty}^{\infty} x^2 f(x)\,\mathrm{d}x - \mu^2

Topic 5: Calculus

SL and HL

SL 5.3
Derivative of xnx^n
f(x)=xn⇒f′(x)=nxn−1f(x) = x^n \Rightarrow f'(x) = nx^{n-1}
SL 5.5
Integral of xnx^n
∫xn dx=xn+1n+1+C\int x^n\,\mathrm{d}x = \frac{x^{n+1}}{n + 1} + C
, n≠−1n \neq -1
SL 5.5
Area between a curve y=f(x)y = f(x) and the xx-axis, where f(x)>0f(x) > 0
A=∫aby dxA = \int_a^b y\,\mathrm{d}x
SL 5.6
Derivative of sin⁡x\sin x
f(x)=sin⁡x⇒f′(x)=cos⁡xf(x) = \sin x \Rightarrow f'(x) = \cos x
SL 5.6
Derivative of cos⁡x\cos x
f(x)=cos⁡x⇒f′(x)=−sin⁡xf(x) = \cos x \Rightarrow f'(x) = -\sin x
SL 5.6
Derivative of ex\mathrm{e}^x
f(x)=ex⇒f′(x)=exf(x) = \mathrm{e}^x \Rightarrow f'(x) = \mathrm{e}^x
SL 5.6
Derivative of ln⁡x\ln x
f(x)=ln⁡x⇒f′(x)=1xf(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}
SL 5.6
Chain rule
y=g(u), where u=f(x)⇒dydx=dydu×dudxy = g(u)\text{, where } u = f(x) \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}
SL 5.6
Product rule
y=uv⇒dydx=udvdx+vdudxy = uv \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}
SL 5.6
Quotient rule
y=uv⇒dydx=vdudx−udvdxv2y = \frac{u}{v} \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}
SL 5.9
Acceleration
a=dvdt=d2sdt2a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}^2 s}{\mathrm{d}t^2}
SL 5.9
Distance travelled from t1t_1 to t2t_2
distance=∫t1t2∣v(t)∣ dt\text{distance} = \int_{t_1}^{t_2} |v(t)|\,\mathrm{d}t
SL 5.9
Displacement from t1t_1 to t2t_2
displacement=∫t1t2v(t) dt\text{displacement} = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t
SL 5.10
Standard integrals
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,\mathrm{d}x = \ln|x| + C
∫sin⁡x dx=−cos⁡x+C\int \sin x\,\mathrm{d}x = -\cos x + C
∫cos⁡x dx=sin⁡x+C\int \cos x\,\mathrm{d}x = \sin x + C
∫ex dx=ex+C\int \mathrm{e}^x\,\mathrm{d}x = \mathrm{e}^x + C
SL 5.11
Area of region enclosed by a curve and xx-axis
A=∫ab∣y∣ dxA = \int_a^b |y|\,\mathrm{d}x

HL only

AHL 5.12
Derivative of f(x)f(x) from first principles
y=f(x)⇒dydx=f′(x)=lim⁡h→0(f(x+h)−f(x)h)y = f(x) \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)
AHL 5.15
Standard derivatives
AHL 5.15
tan⁡x\tan x
f(x)=tan⁡x⇒f′(x)=sec⁡2xf(x) = \tan x \Rightarrow f'(x) = \sec^2 x
AHL 5.15
sec⁡x\sec x
f(x)=sec⁡x⇒f′(x)=sec⁡xtan⁡xf(x) = \sec x \Rightarrow f'(x) = \sec x\tan x
AHL 5.15
cosec⁡x\operatorname{cosec} x
f(x)=cosec⁡x⇒f′(x)=−cosec⁡xcot⁡xf(x) = \operatorname{cosec} x \Rightarrow f'(x) = -\operatorname{cosec} x\cot x
AHL 5.15
cot⁡x\cot x
f(x)=cot⁡x⇒f′(x)=−cosec⁡2xf(x) = \cot x \Rightarrow f'(x) = -\operatorname{cosec}^2 x
AHL 5.15
axa^x
f(x)=ax⇒f′(x)=ax(ln⁡a)f(x) = a^x \Rightarrow f'(x) = a^x(\ln a)
AHL 5.15
log⁡ax\log_a x
f(x)=log⁡ax⇒f′(x)=1xln⁡af(x) = \log_a x \Rightarrow f'(x) = \frac{1}{x\ln a}
AHL 5.15
arcsin⁡x\arcsin x
f(x)=arcsin⁡x⇒f′(x)=11−x2f(x) = \arcsin x \Rightarrow f'(x) = \frac{1}{\sqrt{1 - x^2}}
AHL 5.15
arccos⁡x\arccos x
f(x)=arccos⁡x⇒f′(x)=−11−x2f(x) = \arccos x \Rightarrow f'(x) = -\frac{1}{\sqrt{1 - x^2}}
AHL 5.15
arctan⁡x\arctan x
f(x)=arctan⁡x⇒f′(x)=11+x2f(x) = \arctan x \Rightarrow f'(x) = \frac{1}{1 + x^2}
AHL 5.15
Standard integrals
∫ax dx=1ln⁡aax+C\int a^x\,\mathrm{d}x = \frac{1}{\ln a}a^x + C
∫1a2+x2 dx=1aarctan⁡(xa)+C\int \frac{1}{a^2 + x^2}\,\mathrm{d}x = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
∫1a2−x2 dx=arcsin⁡(xa)+C\int \frac{1}{\sqrt{a^2 - x^2}}\,\mathrm{d}x = \arcsin\left(\frac{x}{a}\right) + C
, ∣x∣<a|x| < a
AHL 5.16
Integration by parts
∫udvdx dx=uv−∫vdudx dx or ∫u dv=uv−∫v du\int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x \text{ or } \int u\,\mathrm{d}v = uv - \int v\,\mathrm{d}u
AHL 5.17
Area of region enclosed by a curve and yy-axis
A=∫ab∣x∣ dyA = \int_a^b |x|\,\mathrm{d}y
AHL 5.17
Volume of revolution about the xx or yy-axes
V=∫abπy2 dx or V=∫abπx2 dyV = \int_a^b \pi y^2\,\mathrm{d}x \text{ or } V = \int_a^b \pi x^2\,\mathrm{d}y
AHL 5.18
Euler’s method
yn+1=yn+h×f(xn,yn); xn+1=xn+hy_{n+1} = y_n + h \times f(x_n, y_n);\ x_{n+1} = x_n + h
where hh is a constant (step length)
AHL 5.18
Integrating factor for y′+P(x)y=Q(x)y' + P(x)y = Q(x)
e∫P(x) dx\mathrm{e}^{\int P(x)\,\mathrm{d}x}
AHL 5.19
Maclaurin series
f(x)=f(0)+xf′(0)+x22!f′′(0)+…f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \ldots
AHL 5.19
Maclaurin series for special functions
ex=1+x+x22!+…\mathrm{e}^x = 1 + x + \frac{x^2}{2!} + \ldots
ln⁡(1+x)=x−x22+x33−…\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots
sin⁡x=x−x33!+x55!−…\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots
cos⁡x=1−x22!+x44!−…\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots
arctan⁡x=x−x33+x55−…\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots

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