IB Mathematics Analysis and Approaches HL Formula Booklet

Prior learning

Description	Formula
$\displaystyle \text{Area of a parallelogram}$	$\displaystyle A = bh, \text{ where } b \text{ is the base, } h \text{ is the height}$
$\displaystyle \text{Area of a triangle}$	$\displaystyle A = \frac{1}{2}(bh), \text{ where } b \text{ is the base, } h \text{ is the height}$
$\displaystyle \text{Area of a trapezoid}$	$\displaystyle A = \frac{1}{2}(a+b)h, \text{ where } a \text{ and } b \text{ are the parallel sides, } h \text{ is the height}$
$\displaystyle \text{Area of a circle}$	$\displaystyle A = \pi r^2, \text{ where } r \text{ is the radius}$
$\displaystyle \text{Circumference of a circle}$	$\displaystyle C = 2\pi r, \text{ where } r \text{ is the radius}$
$\displaystyle \text{Volume of a cuboid}$	$\displaystyle V = lwh, \text{ where } l \text{ is the length, } w \text{ is the width, } h \text{ is the height}$
$\displaystyle \text{Volume of a cylinder}$	$\displaystyle V = \pi r^2h, \text{ where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{Volume of a prism}$	$\displaystyle V = Ah, \text{ where } A \text{ is the area of cross-section, } h \text{ is the height}$
$\displaystyle \text{Area of the curved surface of}\\\text{a cylinder}$	$\displaystyle A = 2\pi rh, \text{ where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{Distance between two}\\\text{points $(x_1, y_1)$ and $(x_2, y_2)$}$	$\displaystyle d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$
$\displaystyle \text{Coordinates of the midpoint of}\\\text{a line segment with endpoints}\\\text{$(x_1, y_1)$ and $(x_2, y_2)$}$	$\displaystyle \displaystyle\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Topic 1: Number and algebra

Topic	Description	Formula
$\displaystyle \text{SL 1.2}$	$\displaystyle \text{The $n$th term of an}\\ \text{arithmetic sequence}$	$\displaystyle u_n = u_1 + (n-1)d$
$\displaystyle \text{SL 1.2}$	$\displaystyle \text{The sum of $n$ terms of an}\\ \text{arithmetic sequence}$	$\displaystyle \displaystyle S_n = \frac{n}{2}(2u_1 + (n-1)d); \; \displaystyle S_n = \frac{n}{2}(u_1 + u_n)$
$\displaystyle \text{SL 1.3}$	$\displaystyle \text{The $n$th term of a}\\ \text{geometric sequence}$	$\displaystyle u_n = u_1r^{n-1}$
$\displaystyle \text{SL 1.3}$	$\displaystyle \text{The sum of $n$ terms of a}\\ \text{finite geometric sequence}$	$\displaystyle \displaystyle S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1-r^n)}{1-r}, \; r \neq 1$
$\displaystyle \text{SL 1.4}$	$\displaystyle \text{Compound interest}$	$\displaystyle \displaystyle FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}, \text{ where }FV\text{ is the future value,}\\[1em] \text{where }PV\text{ is the present value, }n\text{ is the number of years,}\\ \text{$k$ is the number of compounding periods per year,}\\ \text{$r\%$ is the nominal annual rate of interest}$
$\displaystyle \text{SL 1.5}$	$\displaystyle \text{Exponents and logarithms}$	$\displaystyle a^x = b \Leftrightarrow x = \log_a b, \text{ where } a > 0, b > 0, a \neq 1$
$\displaystyle \text{SL 1.7}$	$\displaystyle \text{Exponents and logarithms}$	$\displaystyle \displaystyle \log_a xy = \log_a x + \log_a y\\[1em] \log_a \frac{x}{y} = \log_a x - \log_a y\\[1em] \log_a x^m = m\log_a x\\[1em] \log_a x = \frac{\log_b x}{\log_b a}$
$\displaystyle \text{SL 1.8}$	$\displaystyle \text{The sum of an infinite}\\ \text{geometric sequence}$	$\displaystyle \displaystyle S_\infty = \frac{u_1}{1-r}, \; \|r\| < 1$
$\displaystyle \text{SL 1.9}$	$\displaystyle \text{Binomial theorem } n \in \mathbb{N}$	$\displaystyle \displaystyle (a + b)^n = a^n + {^n\!C_1}a^{n-1}b + ... + {^n\!C_r}a^{n-r}b^r + ... + b^n\\[1em] {^n\!C_r} = \frac{n!}{r!(n-r)!}$
$\displaystyle \text{AHL 1.10}$	$\displaystyle \text{Combinations}$	$\displaystyle \displaystyle {^n\!C_r} = \frac{n!}{r!(n-r)!}$
$\displaystyle \text{AHL 1.10}$	$\displaystyle \text{Permutations}$	$\displaystyle \displaystyle {^n\!P_r} = \frac{n!}{(n-r)!}$
$\displaystyle \text{AHL 1.10}$	$\displaystyle \text{Extension of binomial}\\ \text{theorem, } n \in \mathbb{Q}$	$\displaystyle \displaystyle (a+b)^n = a^n\left(1 + n\left(\frac{b}{a}\right) + \frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^2 + ...\right)$
$\displaystyle \text{AHL 1.12}$	$\displaystyle \text{Complex numbers}$	$\displaystyle z = a + bi$
$\displaystyle \text{AHL 1.13}$	$\displaystyle \text{Modulus-argument (polar)}\\ \text{and exponential (Euler)}\\ \text{form}$	$\displaystyle z = r(\cos \theta + i\sin \theta) = re^{i\theta} = r\text{ cis }\theta$
$\displaystyle \text{AHL 1.14}$	$\displaystyle \text{De Moivre's theorem}$	$\displaystyle [r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta) = r^ne^{in\theta} = r^n\text{ cis }n\theta$

Topic 2: Functions

Topic	Description	Formula
$\displaystyle \text{SL 2.1}$	$\displaystyle \text{Equations of a straight line}$	$\displaystyle y = mx + c; \; ax + by + d = 0; \\ y - y_1 = m(x - x_1)$
$\displaystyle \text{SL 2.1}$	$\displaystyle \text{Gradient formula}$	$\displaystyle \displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
$\displaystyle \text{SL 2.6}$	$\displaystyle \text{Axis of symmetry of the}\\ \text{graph of a quadratic}\\ \text{function}$	$\displaystyle \displaystyle f(x) = ax^2 + bx + c \Rightarrow \text{ axis of symmetry is } x = -\frac{b}{2a}$
$\displaystyle \text{SL 2.7}$	$\displaystyle \text{Solutions of a quadratic}\\ \text{equation}$	$\displaystyle ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \text{ } a \neq 0$
$\displaystyle \text{SL 2.7}$	$\displaystyle \text{Discriminant}$	$\displaystyle \Delta = b^2 - 4ac$
$\displaystyle \text{SL 2.9}$	$\displaystyle \text{Exponential and}\\ \text{logarithmic functions}$	$\displaystyle a^x = e^{x\ln a}; \; \log_a x = \frac{\ln x}{\ln a}, \text{ where } a, x > 0, \; a \neq 1$
$\displaystyle \text{AHL 2.12}$	$\displaystyle \text{Sum and product of the}\\ \text{roots of polynomial}\\ \text{equations of the form}\\ \sum\limits_{r=0}^n a_rx^r = 0$	$\displaystyle \text{Sum is } \frac{-a_{n-1}}{a_n} \; \text{; product is } \frac{(-1)^n a_0}{a_n}$

Topic 3: Geometry and trigonometry

Topic	Description	Formula
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Distance between two}\\\text{points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$}$	$\displaystyle d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Coordinates of the}\\ \text{midpoint of a line segment}\\ \text{with endpoints $(x_1, y_1, z_1)$} \\ \text{and $(x_2, y_2, z_2)$}$	$\displaystyle \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a right-pyramid}$	$\displaystyle V = \frac{1}{3}Ah \text{, where } A \text{ is the area of the base, } h \text{ is the height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a right cone}$	$\displaystyle V = \frac{1}{3}\pi r^2h \text{, where } r \text{ is the radius, } h \text{ is the height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Area of the curved surface}\\\text{of a cone}$	$\displaystyle A = \pi rl \text{, where } r \text{ is the radius, } l \text{ is the slant height}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Volume of a sphere}$	$\displaystyle V = \frac{4}{3}\pi r^3 \text{, where } r \text{ is the radius}$
$\displaystyle \text{SL 3.1}$	$\displaystyle \text{Surface area of a sphere}$	$\displaystyle A = 4\pi r^2 \text{, where } r \text{ is the radius}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Sine rule}$	$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Cosine rule}$	$\displaystyle c^2 = a^2 + b^2 - 2ab\cos C ; \; \cos C = \frac{a^2 + b^2 - c^2}{2ab}$
$\displaystyle \text{SL 3.2}$	$\displaystyle \text{Area of a triangle}$	$\displaystyle A = \frac{1}{2}ab\sin C$
$\displaystyle \text{SL 3.4}$	$\displaystyle \text{Length of an arc}$	$\displaystyle l = r\theta\\ \text{where } r \text{ is the radius, } \theta \text{ is the angle measured in radians}$
$\displaystyle \text{SL 3.4}$	$\displaystyle \text{Area of a sector}$	$\displaystyle \displaystyle A = \frac{1}{2}r^2\theta\\ \text{where } r \text{ is the radius, } \theta \text{ is the angle measured in radians}$
$\displaystyle \text{SL 3.5}$	$\displaystyle \text{Identity for } \tan \theta$	$\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}$
$\displaystyle \text{SL 3.6}$	$\displaystyle \text{Pythagorean identity}$	$\displaystyle \cos^2 \theta + \sin^2 \theta = 1$
$\displaystyle \text{SL 3.6}$	$\displaystyle \text{Double angle identities}$	$\displaystyle \sin 2\theta = 2\sin \theta \cos \theta\\[1em] \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
$\displaystyle \text{AHL 3.9}$	$\displaystyle \text{Reciprocal trigonometric}\\ \text{identities}$	$\displaystyle \sec \theta = \frac{1}{\cos \theta}\\[1em] \cosec \theta = \frac{1}{\sin \theta}$
$\displaystyle \text{AHL 3.9}$	$\displaystyle \text{Pythagorean identities}$	$\displaystyle 1 + \tan^2 \theta = \sec^2 \theta\\[1em] 1 + \cot^2 \theta = \cosec^2 \theta$
$\displaystyle \text{AHL 3.10}$	$\displaystyle \text{Compound angle identities}$	$\displaystyle \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B\\[1em] \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B\\[1em] \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
$\displaystyle \text{AHL 3.10}$	$\displaystyle \text{Double angle identity}\\ \text{for }\tan$	$\displaystyle \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$
$\displaystyle \text{AHL 3.12}$	$\displaystyle \text{Magnitude of a vector}$	$\displaystyle \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Scalar product}$	$\displaystyle \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[1em] \mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\|\|\mathbf{w}\|\cos \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}$
$\displaystyle \text{AHL 3.13}$	$\displaystyle \text{Angle between two}\\ \text{vectors}$	$\displaystyle \cos \theta = \frac{v_1w_1 + v_2w_2 + v_3w_3}{\|\mathbf{v}\|\|\mathbf{w}\|}$
$\displaystyle \text{AHL 3.14}$	$\displaystyle \text{Vector equation of a line}$	$\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$
$\displaystyle \text{AHL 3.14}$	$\displaystyle \text{Parametric form of the}\\ \text{equation of a line}$	$\displaystyle x = x_0 + \lambda l, \text{ } y = y_0 + \lambda m, \text{ } z = z_0 + \lambda n$
$\displaystyle \text{AHL 3.14}$	$\displaystyle \text{Cartesian equations of a}\\ \text{line}$	$\displaystyle \frac{x-x_0}{l} = \frac{y-y_0}{m} = \frac{z-z_0}{n}$
$\displaystyle \text{AHL 3.16}$	$\displaystyle \text{Vector product}$	$\displaystyle \mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}, \text{ where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix}\\[2em] \|\mathbf{v} \times \mathbf{w}\| = \|\mathbf{v}\|\|\mathbf{w}\|\sin \theta, \text{ where } \theta \text{ is the angle between } \mathbf{v} \text{ and } \mathbf{w}$
$\displaystyle \text{AHL 3.16}$	$\displaystyle \text{Area of a parallelogram}$	$\displaystyle A = \|\mathbf{v} \times \mathbf{w}\| \text{ where } \mathbf{v} \text{ and } \mathbf{w} \text{ form two adjacent sides of a}\\ \text{parallelogram}$
$\displaystyle \text{AHL 3.17}$	$\displaystyle \text{Vector equation of a plane}$	$\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$
$\displaystyle \text{AHL 3.17}$	$\displaystyle \text{Equation of a plane}\\ \text{(using the normal vector)}$	$\displaystyle \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$
$\displaystyle \text{AHL 3.17}$	$\displaystyle \text{Cartesian equation of a}\\ \text{plane}$	$\displaystyle ax + by + cz = d$

Topic 4: Statistics and probability

Topic	Description	Formula
$\displaystyle \text{SL 4.2}$	$\displaystyle \text{Interquartile range}$	$\displaystyle \operatorname{IQR} = Q_3 - Q_1$
$\displaystyle \text{SL 4.3}$	$\displaystyle \text{Mean, } \bar{x} \text{, of a set of data}$	$\displaystyle \bar{x} = \frac{\sum\limits_{i=1}^k f_ix_i}{n}, \text{ where } n = \sum\limits_{i=1}^k f_i$
$\displaystyle \text{SL 4.5}$	$\displaystyle \text{Probability of an event } A$	$\displaystyle \operatorname{P}(A) = \frac{n(A)}{n(U)}$
$\displaystyle \text{SL 4.5}$	$\displaystyle \text{Complementary events}$	$\displaystyle \operatorname{P}(A) + \operatorname{P}(A') = 1$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Combined events}$	$\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B) - \operatorname{P}(A \cap B)$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Mutually exclusive events}$	$\displaystyle \operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B)$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Conditional probability}$	$\displaystyle \operatorname{P}(A\|B) = \frac{\operatorname{P}(A \cap B)}{\operatorname{P}(B)}$
$\displaystyle \text{SL 4.6}$	$\displaystyle \text{Independent events}$	$\displaystyle \operatorname{P}(A \cap B) = \operatorname{P}(A)\operatorname{P}(B)$
$\displaystyle \text{SL 4.7}$	$\displaystyle \text{Expected value of a}\\ \text{discrete random variable } X$	$\displaystyle \operatorname{E}(X) = \sum x\operatorname{P}(X = x)$
$\displaystyle \text{SL 4.8}$	$\displaystyle \text{Binomial distribution}\\ X \sim \text{B}(n, p)$	$\displaystyle \operatorname{E}(X) = np$
$\displaystyle \text{SL 4.8}$	$\displaystyle \text{Mean}$	$\displaystyle \operatorname{E}(X) = np$
$\displaystyle \text{SL 4.8}$	$\displaystyle \text{Variance}$	$\displaystyle \operatorname{Var}(X) = np(1-p)$
$\displaystyle \text{SL 4.12}$	$\displaystyle \text{Standardized normal}\\ \text{variable}$	$\displaystyle z = \frac{x - \mu}{\sigma}$
$\displaystyle \text{AHL 4.13}$	$\displaystyle \text{Bayes' theorem}$	$\displaystyle \operatorname{P}(B\|A) = \frac{\operatorname{P}(B)\operatorname{P}(A\|B)}{\operatorname{P}(B)\operatorname{P}(A\|B) + \operatorname{P}(B')\operatorname{P}(A\|B')}\\[2em] \operatorname{P}(B_i\|A) = \frac{\operatorname{P}(B_i)\operatorname{P}(A\|B_i)}{\operatorname{P}(B_1)\operatorname{P}(A\|B_1) + \operatorname{P}(B_2)\operatorname{P}(A\|B_2) + \operatorname{P}(B_3)\operatorname{P}(A\|B_3)}$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Variance } \sigma^2$	$\displaystyle \sigma^2 = \frac{\sum\limits_{i=1}^k f_i(x_i - \mu)^2}{n} = \frac{\sum\limits_{i=1}^k f_ix_i^2}{n} - \mu^2$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Standard deviation } \sigma$	$\displaystyle \sigma = \sqrt{\frac{\sum\limits_{i=1}^k f_i(x_i - \mu)^2}{n}}$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Linear transformation of a}\\ \text{single random variable}$	$\displaystyle \operatorname{E}(aX + b) = a\operatorname{E}(X) + b\\[1em] \operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Expected value of a}\\ \text{continuous random}\\ \text{variable } X$	$\displaystyle \operatorname{E}(X) = \mu = \int_{-\infty}^{\infty} xf(x)\,dx$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Variance}$	$\displaystyle \operatorname{Var}(X) = \operatorname{E}[(X-\mu)^2] = \operatorname{E}(X^2) - [\operatorname{E}(X)]^2$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Variance of a discrete}\\ \text{random variable } X$	$\displaystyle \operatorname{Var}(X) = \sum(x-\mu)^2\operatorname{P}(X=x) = \sum x^2\operatorname{P}(X=x) - \mu^2$
$\displaystyle \text{AHL 4.14}$	$\displaystyle \text{Variance of a continuous}\\ \text{random variable } X$	$\displaystyle \operatorname{Var}(X) = \int_{-\infty}^{\infty}(x-\mu)^2f(x)\,dx = \int_{-\infty}^{\infty}x^2f(x)\,dx - \mu^2$

Topic 5: Calculus

Topic	Description	Formula
$\displaystyle \text{SL 5.3}$	$\displaystyle \text{Derivative of } x^n$	$\displaystyle f(x) = x^n \Rightarrow f'(x) = nx^{n-1}$
$\displaystyle \text{SL 5.5}$	$\displaystyle \text{Integral of } x^n$	$\displaystyle \int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \text{ } n \neq -1$
$\displaystyle \text{SL 5.5}$	$\displaystyle \text{Area between a curve}\\ \text{$y = f(x)$ and the $x$-axis,}\\ \text{where } f(x) > 0$	$\displaystyle A = \int_a^b y\,dx$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Derivative of } \sin x$	$\displaystyle f(x) = \sin x \Rightarrow f'(x) = \cos x$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Derivative of } \cos x$	$\displaystyle f(x) = \cos x \Rightarrow f'(x) = -\sin x$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Derivative of } e^x$	$\displaystyle f(x) = e^x \Rightarrow f'(x) = e^x$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Derivative of } \ln x$	$\displaystyle f(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Chain rule}$	$\displaystyle y = g(u), \text{ where } u = f(x) \Rightarrow \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Product rule}$	$\displaystyle y = uv \Rightarrow \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$
$\displaystyle \text{SL 5.6}$	$\displaystyle \text{Quotient rule}$	$\displaystyle y = \frac{u}{v} \Rightarrow \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$
$\displaystyle \text{SL 5.9}$	$\displaystyle \text{Acceleration}$	$\displaystyle a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
$\displaystyle \text{SL 5.9}$	$\displaystyle \text{Distance travelled from}\\ \text{$t_1$ to $t_2$}$	$\displaystyle \text{distance} = \int_{t_1}^{t_2} \|v(t)\|\,dt$
$\displaystyle \text{SL 5.9}$	$\displaystyle \text{Displacement from}\\ \text{$t_1$ to $t_2$}$	$\displaystyle \text{displacement} = \int_{t_1}^{t_2} v(t)\,dt$
$\displaystyle \text{SL 5.10}$	$\displaystyle \text{Standard integrals}$	$\displaystyle \int \frac{1}{x}\,dx = \ln\|x\| + C\\[1em] \int \sin x\,dx = -\cos x + C\\[1em] \int \cos x\,dx = \sin x + C\\[1em] \int e^x\,dx = e^x + C$
$\displaystyle \text{SL 5.11}$	$\displaystyle \text{Area of region enclosed}\\ \text{by a curve and $x$-axis}$	$\displaystyle A = \int_a^b \|y\|\,dx$
$\displaystyle \text{AHL 5.12}$	$\displaystyle \text{Derivative of } f(x) \text{ from}\\ \text{first principles}$	$\displaystyle y = f(x) \Rightarrow \frac{dy}{dx} = f'(x) = \lim_{h \to 0}\left(\frac{f(x+h) - f(x)}{h}\right)$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \text{Standard derivatives}\\ \text{tan }x$	$\displaystyle f(x) = \tan x \Rightarrow f'(x) = \sec^2 x$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \sec x$	$\displaystyle f(x) = \sec x \Rightarrow f'(x) = \sec x\tan x$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \cosec x$	$\displaystyle f(x) = \cosec x \Rightarrow f'(x) = -\cosec x\cot x$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \cot x$	$\displaystyle f(x) = \cot x \Rightarrow f'(x) = -\cosec^2 x$
$\displaystyle \text{AHL 5.15}$	$\displaystyle a^x$	$\displaystyle f(x) = a^x \Rightarrow f'(x) = a^x(\ln a)$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \log_a x$	$\displaystyle f(x) = \log_a x \Rightarrow f'(x) = \frac{1}{x\ln a}$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \arcsin x$	$\displaystyle f(x) = \arcsin x \Rightarrow f'(x) = \frac{1}{\sqrt{1-x^2}}$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \arccos x$	$\displaystyle f(x) = \arccos x \Rightarrow f'(x) = -\frac{1}{\sqrt{1-x^2}}$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \arctan x$	$\displaystyle f(x) = \arctan x \Rightarrow f'(x) = \frac{1}{1+x^2}$
$\displaystyle \text{AHL 5.15}$	$\displaystyle \text{Standard integrals}$	$\displaystyle \int a^x\,dx = \frac{1}{\ln a}a^x + C\\[1em] \int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C\\[1em] \int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C, \text{ }\|x\| < a$
$\displaystyle \text{AHL 5.16}$	$\displaystyle \text{Integration by parts}$	$\displaystyle \int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx \text{ or } \int u\,dv = uv - \int v\,du$
$\displaystyle \text{AHL 5.17}$	$\displaystyle \text{Area of region enclosed}\\ \text{by a curve and $y$-axis}$	$\displaystyle A = \int_c^d \|x\|\,dy$
$\displaystyle \text{AHL 5.17}$	$\displaystyle \text{Volume of revolution}\\ \text{about the $x$ or $y$-axes}$	$\displaystyle V = \int_a^b \pi y^2\,dx \text{ or } V = \int_c^d \pi x^2\,dy$
$\displaystyle \text{AHL 5.18}$	$\displaystyle \text{Euler's method}$	$\displaystyle y_{n+1} = y_n + h \times f(x_n, y_n); \; x_{n+1} = x_n + h, \text{ where $h$ is a constant}\\ \text{(step length)}$
$\displaystyle \text{AHL 5.18}$	$\displaystyle \text{Integrating factor for}\\ y' + P(x)y = Q(x)$	$\displaystyle e^{\int P(x)\,dx}$
$\displaystyle \text{AHL 5.19}$	$\displaystyle \text{Maclaurin series}$	$\displaystyle f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + ...$
$\displaystyle \text{AHL 5.19}$	$\displaystyle \text{Maclaurin series for}\\ \text{special functions}$	$\displaystyle e^x = 1 + x + \frac{x^2}{2!} + ...\\[1em] \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...\\[1em] \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...\\[1em] \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...\\[1em] \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - ...$

Analysis and Approaches HLFormula Booklet

Prior learning

Topic 1: Number and algebra

Topic 2: Functions

Topic 3: Geometry and trigonometry

Topic 4: Statistics and probability

Topic 5: Calculus