List of math formulas
Here you can find a summary of the main formulas you need to know. This list was not organized by years of schooling but thematically. Just choose one of the topics and you will be able to view the formulas related to this subject. This is not an exhaustive list, ie it's not here all math formulas that are used in mathematics class, only those that were considered most important.
| Square |  | `A=l^2` | `l` : length of side |
| Rectangle |  | `A=wxxh` | `w` : width `h` : height |
| Triangle |  | `A=(bxxh)/2` | `b` : base `h` : height |
| Rhombus |  | `A=(Dxxd)/2` | `D` : large diagonal `d` : small diagonal |
| Trapezoid |  | `A=(B+b)/2xxh` | `B` : large side `b` : small side `h`: height |
| Regular polygon |  | `A=P/2xxa` | `P` : perimeter `a` : apothem |
| Circle |  | `A=pir^2` `P=2pir` | `r` : radius `P` : perimeter |
| Cone (lateral surface) |  | `A=pirxxs` | `r` : radius `s` : slant height |
| Sphere (surface area) |  | `A=4pir^2` | `r`: radius |
| Cube |  | `V=s^3` | `s`: side |
| Parallelepiped |  | `V=lxxwxxh` | `l`: length `w`: width `h`: height |
| Regular prism |  | `V=bxxh` | `b`: base `h`: height |
| Cylinder |  | `V=pir^2xxh` | `r`: radius `h`: height |
| Cone (or pyramid) | | `V=1/3bxxh` | `b`: base `h`: height |
| Sphere | | `V=4/3pir^3` | `r`: radius |
| Directly Proportional | `y = kx` `k = y/x` | `k`: Constant of Proportionality |
| Inversely Proportional | `y = k/x` `k = yx` | |
| `ax^2+bx+c=0` | Quadratic formula | `x=(-b +- sqrt(b^2 - 4ac))/(2a)` |
| Concavity | Concave up: `a > 0` | |
| Concave down: `a < 0` | ||
| Discriminant | `Delta = b^2 - 4ac` | |
| Vertex of the parabola | `V((-b)/(2a),(-Delta)/(4a))` | |
| `y=a(x-h)^2+k` | Concavity | Concave up: `a > 0` |
| Concave down: `a < 0` | ||
| Vertex of the parabola | `V(h, k)` | |
| Zero-product property | `AxxB=0 hArr A=0 vv B=0` | ex : `(x+2)xx(x-1)=0 hArr ` `x+2=0 vv x-1=0 hArr x=-2 vv x=1` |
| Difference of two squares | `(a-b)(a+b)=a^2 - b^2` | ex : `(x-2)(x+2)=x^2 - 2^2=x^2 - 4` |
| Perfect square trinomial | `(a+b)^2=a^2 + 2ab + b^2` | ex : `(2x+3)^2=(2x)^2 + 2*2x*3 +3^2=` `4x^2 + 12x + 9` |
| Binomial theorem | `(x + y)^n = sum_(k=0)^n text( )^nC_k text( ) x^(n-k) text( ) y^k` | |
| Product | `a^mxxa^n=a^(m+n)` | ex : `3^5xx3^2=3^(5+2)=3^7` |
| `a^mxxb^m=(axxb)^m` | ex : `3^5xx2^5=(3xx2)^5=6^5` | |
| Quotient | `a^m-:a^n=a^(m-n)` | ex : `3^7-:3^2=3^(7-2)=3^5` |
| `a^m-:b^m=(a-:b)^m` | ex : `6^5-:2^5=(6-:2)^5=3^5` ex : `5^3-:2^3=(5/2)^3` | |
| Power of Power | `(a^m)^p=a^(mxxp)` | ex : `(5^2)^3=5^(2xx3)=5^6` |
| Zero Exponents | `a^0=1` | ex : `8^0=1` |
| Negative Exponents | `a^-n=(1/a)^n` | ex : `3^-2=(1/3)^2` ex : `(2/3)^-4=(3/2)^4` |
| Fractional Exponents | `a^(p/q)=root(q)(a^p)` | ex : `2^(4/3) = root(3)(2^4)` |
| Multiplication | `root(n)(x)xxroot(n)(y)=root(n)(x xx y)` | ex : `root(3)(2)xxroot(3)(5)=root(3)(2xx5) hArr root(3)(10)` |
| Division | `root(n)(x)-:root(n)(y)=root(n)(x/y)` | ex : `root(4)(8)-:root(4)(3)=root(4)(8/3)` |
| Addition | `a root(n)(x)+-b root(n)(x)=(a+-b)root(n)(x)` | ex : `4root(3)(5)-2root(3)(5)=(4-2)root(3)(5) hArr 2root(3)(5)` |
| Exponents | `(root(n)(x))^p=root(n)(x^p)` | ex : `(sqrt 2)^3=sqrt (2^3) hArr sqrt 8` |
| Radicals | `root(n)(root(p)(x))=root(n*p)(x)` | ex : `root(3)(sqrt 5)=root (3xx2)(5) hArr root(6)(5)` |
| Exponentiation | `root(n)(a^m)=a^(m/n)` | ex : `root(3)(4^5)=4^(5/3)` |
| Simplifying Radicals | `(root(n)(a))^n=a` | ex : `(sqrt(3))^2=3` |
| `(root(n)(a))^m=root(n)(a^m)` | ex : `(sqrt(4))^5=sqrt(4^5)` |
| Trigonometry Ratios |  | `sin alpha=(opp.)/ (hip.)` | `opp.`: opposite `hip.`: hypotenuse |
| `cos alpha=(adj.)/(hip.)` | `adj.`: adjacent `hip.`: hypotenuse | ||
| `tan alpha=(opp.)/(adj.)` | `opp.`: opposite `adj.`: adjacent | ||
| Fundamental Identities | `sin^2 alpha + cos^2 alpha=1` | `tan alpha=(sin alpha)/(cos alpha)` | `tan^2 alpha + 1 = 1/(cos^2 alpha)` |
 | Law of Sines (aka sine rule) | `(sin A)/a = (sin B)/b = (sin C)/c` | |
| Law of Cosines (aka cosine rule) | `a^2=b^2+c^2-2bc cos A` | ||
| Heron's formula | `A=sqrt(s(s-a)(s-b)(s-c))` `s=(a+b+c)/2` | ||
| Exact Values | `sin (pi/6)=1/2` | `cos (pi/6)=sqrt(3)/2` | `tan (pi/6)=sqrt(3)/3` |
| `sin (pi/4)=sqrt(2)/2` | `cos (pi/4)=sqrt(2)/2` | `tan (pi/4)=1` | |
| `sin (pi/3)=sqrt(3)/2` | `cos (pi/3)=1/2` | `tan (pi/3)=sqrt(3)` | |
| Angle Relationships | `sin (-alpha)=-sin alpha` | `cos (- alpha)=cos alpha` | `tan (-alpha)=-tan alpha` |
| `sin (pi - alpha)=sin alpha` | `cos (pi - alpha)=-cos alpha` | `tan (pi - alpha)=-tan alpha` | |
| `sin (pi + alpha)=-sin alpha` | `cos (pi + alpha)=-cos alpha` | `tan (pi + alpha)=tan alpha` | |
| `sin (pi/2 - alpha)=cos alpha` | `cos (pi/2 - alpha)=sin alpha` | `tan (pi/2 - alpha)=1/(tan alpha)` | |
| `sin (pi/2 + alpha)=cos alpha` | `cos (pi/2 + alpha)=-sin alpha` | `tan (pi/2 + alpha)=-1/(tan alpha)` | |
| `sin ((3pi)/2 - alpha)=-cos alpha` | `cos ((3pi)/2 - alpha)=-sin alpha` | `tan ((3pi)/2 - alpha)=1/(tan alpha)` | |
| `sin ((3pi)/2 + alpha)=-cos alpha` | `cos ((3pi)/2 + alpha)=sin alpha` | `tan ((3pi)/2 + alpha)=-1/(tan alpha)` | |
| Trigonometric Equations | `sin x=sin alpha hArr x = alpha + 2kpi vv x = pi - alpha + 2kpi, k in ZZ ` | ||
| `cos x=cos alpha hArr x = alpha + 2kpi vv x = - alpha + 2kpi, k in ZZ ` | |||
| `tan x=tan alpha hArr x = alpha + kpi, k in ZZ ` | |||
| Sum Formulas | `sin (a+b)=sin a xx cos b + sin b xx cos a` | ||
| `cos (a+b)=cos a xx cos b - sin a xx sin b` | |||
| `tan (a+b)=(tan a + tan b) / (1 - tan a xx tan b)` | |||
| Difference Formulas | `sin (a-b)=sin a xx cos b - sin b xx cos a` | ||
| `cos (a-b)=cos a xx cos b + sin a xx sin b` | |||
| `tan (a-b)=(tan a - tan b) / (1 + tan a xx tan b)` | |||
| Double Angle Formulas | `sin (2a)=2xxsin a xx cos a` | ||
| `cos (2a)=cos ^2 a - sin^2 a` | |||
| `tan (2a)=(2 xx tan a) / (1 - tan^2 a)` | |||
| Euler's Polyhedral Formula | `F + V = E + 2` | `F`: Face `V`: Vertex `E`: Edge |
| Sum of interior angles of a polygon | `S_i=(n-2)xx180º` | `n`: Number of sides |
| Pythagorean theorem | `H^2=C_1^2+C_2^2` | Hypotenuse: `H` Leg: `C_1` e `C_2` |
| Distance between two points | `bar (AB)=sqrt((x_1-x_2)^2+(y_1-y_2)^2)` | ex: `A(8,2)` e `B(4,-1)` `bar (AB)=sqrt((8-4)^2+(2+1)^2) hArr` `bar(AB)=sqrt(16+9) hArr bar(AB)=5` |
| Midpoints | `M((x_1+x_2)/2,(y_1+y_2)/2)` | ex: `A(2,6)` e `B(4,-2)` `M((2+4)/2,(6-2)/2) hArr M(3,2)` |
| Equation of a straight line | Slope–intercept form Slope: `m`, Y intercept: `b` | `y=mx+b` |
| Vector Form Direction vector: `vec u(u_1,u_2,u_3)` Point`(x_0,y_0,z_0)` | `(x,y,z)=(x_0,y_0,z_0)+k(u_1,u_2,u_3), k in RR` | |
| Cartesian Form Direction vector: `vec u(u_1,u_2,u_3)` Point`(x_0,y_0,z_0)` | `(x - x_0)/u_1=(y - y_0)/u_2=(z - z_0)/u_3` | |
| Parametric Form Direction vector: `vec u(u_1,u_2,u_3)` Point`(x_0,y_0,z_0)` | `{(x = x_0 + Ku_1),(y = y_0 + Ku_2),(z = z_0 + Ku_3):}, k in RR` | |
| Equation of a plane | Cartesian Form Normal vector: `vec u(n_1,n_2,n_3)` Point`(x_0,y_0,z_0)` | `n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0` |
| Scalar Form Normal vector: `vec u(n_1,n_2,n_3)` | `n_1x + n_2y + n_3z +d = 0` | |
| Equation of a circle | Center `(x_0,y_0)` and radius `r` | `(x-x_0)^2+(y-y_0)^2=r^2` |
| Equation of a Sphere | Center `(x_0,y_0,z_0)` and radius `r` | `(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2` |
| Equation of an Ellipse | Center `(h, k)` Axis `a` and `b` | `((x-h)/a)^2+((y-k)/b)^2=1` |
| Conjunction | Disjunction | Implication | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| `p` | `q` | `p ^^ q` | `p` | `q` | `p vv q` | `p` | `q` | `p rArr q` | |||
| V | V | V | V | V | V | V | V | V | |||
| V | F | F | V | F | V | V | F | F | |||
| F | V | F | F | V | V | F | V | V | |||
| F | F | F | F | F | F | F | F | V | |||
| Law of noncontradiction | `p ^^ ~p hArr F` | |
| Law of the excluded middle | `p vv ~p hArr V` | |
| Double Negation | `~(~p) hArr p` | |
| Commutativity | Conjunction | `p ^^ q hArr q ^^ p` |
| Disjunction | `p vv q hArr q vv p` | |
| Associativity | Conjunction | `(p ^^ q) ^^ r hArr p ^^ (q ^^ r)` |
| Disjunction | `(p vv q) vv r hArr p vv (q vv r)` | |
| Neutral Element | Conjunction | `p ^^ V hArr p` |
| Disjunction | `p vv F hArr p` | |
| Absorbing Element | Conjunction | `p ^^ F hArr F` |
| Disjunction | `p vv V hArr V` | |
| Idempotence | Conjunction | `p ^^ p hArr p` |
| Disjunction | `p vv p hArr p` | |
| Distributive Property | Conjunction over Disjunction | `p ^^ (q vv r) hArr (p ^^ q) vv (p ^^ r)` |
| Disjunction over Conjunction | `p vv (q ^^ r) hArr (p vv q) ^^ (p vv r)` | |
| Properties of Implication | Transitive | `(p rArr q) ^^ (q rArr r) rArr (p rArr r)` |
| Implication and Disjunction | `(p rArr q) hArr ~p vv q` | |
| Negation | `~(p rArr q) hArr p ^^ ~q` | |
| Contrapositive of an Implication | `(p rArr q) hArr (~q rArr ~p)` | |
| Properties of Equivalence | Double implication | `(p hArr q) hArr [(p rArr q) ^^ (q rArr p)]` |
| Transitive | `[(p hArr q) ^^ (q hArr r)] rArr (p hArr r)` | |
| Negation | `~(p hArr q) hArr [(p ^^ ~q) vv (q ^^ ~p)]` | |
| De Morgan's laws | Negation of a Conjunction | `~(p ^^ q) hArr ~p vv ~q` |
| Negation of a Disjunction | `~(p vv q) hArr ~p ^^ ~q` | |
| De Morgan's laws | Negation of Universal Quantifier | `~(AAx, p(x)) hArr EEx: ~p(x)` |
| Negation of Existential Quantifier | `~(EEx: p(x)) hArr AAx, ~p(x)` | |
| Notation | `vec(AB)=B - A = (b_1-a_1,b_2-a_2)` | ex : `A(3,2)` and `B(4,5)` `vec(AB)=(4,5)-(3,2)=(4-3,5-2)=(1,3)` | |
| Magnitude | `||vec u||=sqrt((u_1)^2 + (u_2)^2)` | ex : `vec u(3,2)` `||vec u||=sqrt(3^2+2^2) hArr ||vec u||=sqrt 13` | |
| Square of magnitude of a vector | `(vec u)^2 = ||vec u||^2` | ex : `vec u(4,3)` and `||vec u||=5` then `(vec u)^2 = 5^2` | |
| Calculations | `A+vec u=(a_1+u_1, a_2+u_2)` | ex : `A(4,5)` and `vec u(3,2)` `A+vec u=(4+3, 5+2) hArr A+vec u=(7, 7)` | |
| `vec u+vec v=(u_1+v_1, u_2+v_2)` | ex : `vec u(6,3)` and `vec v(2,1)` `vec u+vec v=(6+2, 3+1) hArr vec u+vec v=(8, 4)` | ||
| `kxxvec u=(kxxu_1, kxxu_2)` | ex : `k=2` and `vec u(3,4)` `kxxvec u=(2xx3, 2xx4) hArr kxxvec u=(6, 8)` | ||
| The Scalar or Dot Product | `vec u.vec v=u_1xxv_1+u_2xxv_2` | ex : `vec u(2,1)` and `vec v(0,3)` `vec u.vec v=2xx0+1xx3` `vec u.vec v=3` | |
| `vec u.vec v=||vec u||xx||vec v||xxcos(vec u \^ vec v)` | |||
| Angle between two lines | Direction vector of lines: `vec u` and `vec v` angle: `alpha` | `cos alpha=|vec u.vec v|/(||vec u||xx||vec v||)` | |
| To use the above concepts in space, just add a third coordinate. | |||
| Summation Rules and Properties | `sum_(i=p)^n lambda = (n-p+1)lambda` | |
| `sum_(i=1)^n lambda x_i = lambda sum_(i=1)^n x_i` | ||
| `sum_(i=1)^n (x_i + y_i) = sum_(i=1)^n x_i + sum_(i=1)^n y_i` | ||
| `sum_(i=1)^n x_i = sum_(i=1)^p x_i + sum_(i=p+1)^n x_i` | ||
| Used Symbols | Statistical sample | `x = (x_1, x_2, x_3, ..., x_n)` |
| Sample size | `N` | |
| Absolute Frequency | `n_i` | |
| Relative Frequency | `f_i = n_i / N` | |
| Cumulative (Absolute) Frequency | `N_i` | |
| Cumulative Relative Frequency | `F_i` | |
| Sample Mean | Ungrouped Data | `bar(x) = (sum_(i=1)^k x_i)/N` |
| Grouped Data | `bar(x) = (sum_(i=1)^k n_i x_i)/N` | |
| `bar(x) = sum_(i=1)^k f_i x_i` | ||
| Median | If N is odd | `Me = x_k, k = (N+1)/2` |
| If N is even | `Me = (x_k + x_(k+1))/2, k = N/2` | |
| Sum of Deviations from the Mean | `sum_(i=1)^k d_i = sum_(i=1)^k (x_i - bar(x)) = 0` | |
| Sum of Squared Deviations from the Mean | Ungrouped Data | `SS_x = sum_(i=1)^k (x_i - bar(x))^2` |
| `SS_x = sum_(i=1)^k x_i^2 - k bar(x)^2` | ||
| Grouped Data | `SS_x = sum_(i=1)^k (x_i - bar(x))^2 n_i` | |
| Sample Variance | `S_x^2 = (SS_x)/(N-1)` | |
| Sample Standard Deviation | `S_x = sqrt((SS_x)/(N-1))` | |
| Arithmetic sequences | Common difference | `r = u_(n+1) - u_n` |
| Expression for the nth term | `u_n=u_1+(n-1)r` | |
| Monotonicity | Increasing if `r>0` Decreasing if `r < 0` | |
| Sum of the first n terms | `S_n=(u_1+u_n)/2xxn` | |
| Geometric sequences | Common ratio | `r = u_(n+1) / u_n` |
| Expression for the nth term | `u_n=u_1xxr^(n-1)` | |
| Monotonicity | Increasing if `u_1>0 ^^ r>1` Decreasing if `u_1 < 0 ^^ r>1` Not Monotonic if `r < 0` | |
| Sum of the first n terms | `S_n=u_1xx(1-r^n)/(1-r)` | |
| Simple Interest | `FV = P xx (1 + r xx t)` | `FV` : Future Value `P` : Principal `t` : time `r` : interest rate |
| Compound Interest | `FV = P xx (1 + r)^t` |
| Average rate of change between two points | Slope of the Secant Line `[a,b]` | `SSL=(f(b)-f(a))/(b-a)` |
| Rate of change at a point | `f'(x_0)=lim_(x->x_0)(f(x)-f(x_0))/(x-x_0)` | `f'(x_0)=lim_(h->0)(f(x_0+h)-f(x_0))/h` |
| Constant | `a'=0` | ex : `4'=0` |
| Multiplication by constant | `(mx)'=m` | ex : `(3x)'=3` |
| Power Rule | `(u^n)'=nxxu^(n-1)xxu'` | ex : `((6x)^5)'=5(6x)^4xx(6x)'=5(6x)^4xx6` |
| Root | `(root(n)(u))'=(u')/(n xx root(n)(u^(n-1)))` | ex : `(sqrt(2x))'=((2x)')/(2 xx sqrt(2x))=1/(sqrt(2x))` |
| Exponential | `(a^u)'=u'xxa^uxxln a` | ex : `(7^(3x))'=3xx7^(3x)xxln7` |
| Exponential base `e` | `(e^u)'=u'xxe^u` | ex : `(e^(2x))'=2xxe^(2x)` |
| Sum Rule | `(u+v)'=u'+v'` | ex : `(2x+5)'=(2x)'+5'=2` |
| Product Rule | `(uxxv)'=u'v + uv'` | ex : `(x^2xxe^x)=(x^2)'e^x+x^2(e^x)'=2xe^x+x^2e^x` |
| Quotient Rule | `(u/v)'=(u'v - uv')/v^2` | ex : `((x+1)/(2x))' = ((x+1)'xx(2x) - (x+1)xx(2x)')/(2x)^2` |
| Chain Rule | `(g o f)'=g'(f) xx f'` | ex : `g(x)=2x^2;g'(x)=4x;f(x)=2x;f'(x)=2` `(gof)'=4(2x)xx2` |
| Sine | `(sin u)'=u'xxcosu` | ex : `(sin(6x))'=6xxcos(6x)` |
| Cosine | `(cos u)'=-u'xxsinu` | ex : `(cos(3x))'=-3xxsin(3x)` |
| Tangent | `(tan u)'=(u')/(cos^2u)` | ex : `(tan(x))'=1/(cos^2x)` |
| Logarithms | `(log_a u)'=(u')/(uxxln a)` | ex : `(log_4 (6x))'=((6x)´)/(6xln 4)=6/(6xln 4)=1/(xln 4)` |
| Natural logarithm | `(ln u)'=(u')/(u)` | ex : `(ln (5x))'=((5x)´)/(5x)=5/(5x)=1/x` |
| Commutative | `A uu B = B uu A` | `A nn B = B nn A` |
| Associative | `A uu (B uu C) = A uu (B uu C)` | `A nn (B nn C) = A nn (B nn C)` |
| Neutral element | `A uu O/ = A` | `A nn E = A` |
| Absorbing element | `A uu E = E` | `A nn O/ = O/` |
| Distributive | `A uu (B nn C) = (A uu B) nn (A uu C)` | `A nn (B uu C) = (A nn B) uu (A nn C)` |
| De Morgan's laws | `bar(A nn B) = bar(A) uu bar(B)` | `bar(A uu B) = bar(A) nn bar(B)` |
| Laplace laws | `P(A) = text(Number of ways it can happen)/text(Total number of outcomes)` | |
| Complement of an Event | `P(bar(A)) = 1 - P(A)` | |
| Union of Events | `P(A uu B) = P(A) + P(B) - P(A nn B)` | |
| Conditional Probability | `P(A | B) = (P(A nn B)) / (P(B))` | |
| Independent Events | `P(A | B) = P(A)` | `P(A nn B) = P(A) xx P(B)` |
| Permutation | `P_n = n! = n xx (n - 1) xx ... xx 2 xx 1` | ex : `P_4 = 4! = 4 xx 3 xx 2 xx 1 = 24` |
| Permutations without repetition | `text()^nA_p = (n!)/((n-p)!)` | ex : `text()^6A_2 = (6!)/((6-2)!)=30` |
| Permutations with repetition | `text()^nA_p^' = n^p` | ex : `text()^5A_3^' = 5^3=125` |
| Combination | `text()^nC_p = (text()^nA_p)/(p!)=(n!)/((n-p)! xx p!)` | ex : `text()^5C_4 = (text()^5A_4)/(4!)=5` |
| Probability Distribution | Average value | `mu = x_1p_1 + x_2p_2 + ... + x_kp_k` |
| Standard deviation | `sigma=sqrt(sum_(i=1)^k p_i(x_i-mu)^2` | |
| Binomial distribution | `P(X=k) = text()^nC_k.p^k.(1-p)^(n-k)` | ex : `B(10;0,6)` `P(X=3) = text()^10C_3xx0,6^3xx0,4^7` |
| Definition | `log_a b = x hArr b=a^x` | ex : `3^x=15 hArr x=log_3 15` |
| `log_a 1 = 0` | ex : `log_3 1 = 0` | |
| `log_a a = 1` | ex : `log 10 = 1` | |
| `log_a a^b = b` | ex : `ln e^2 = 2` | |
| Product | `log_a (uxxv) = log_a u + log_a v` | ex : `log_6 10 + log_6 2 = log_6 (10xx2) = log_6 20` |
| Quotient | `log_a (u/v) = log_a u - log_a v` | ex : `log_4 9 - log_4 3 = log_4 (9/3) = log_4 3` |
| Exponential | `log_a u^v = vxxlog_a u` | ex : `log_4 36 = log_4 6^2= 2xxlog_4 6` |
| Change of Base | `log_a u = (log_b u)/(log_b a)` | ex : `log_4 5 xx log_5 6 = log_4 5 xx (log_4 6)/(log_4 5) = log_4 6` |
| `lim_(x->+oo) a^x/x^p = +oo` `(a, p in RR)` | `lim_(x->+oo) (log_a x) / x = 0` `(a > 1, a in RR)` |
| `lim_(x->0) (e^x - 1)/x = 1` | `lim_(x->0) (ln (x+1)) / x = 1` |
| `lim_(x->0) sin x/x = 1` | `lim_(x->+oo) sin x/x = 0` |
| `lim_(u_n->+oo)(1 + k/(u_n))^(u_n) = e^k` | `lim (1 + 1/n)^n = e` `(n in NN)` |
| Common primitives | `int 1` `dx = x + c, c in RR` |
| `int (u(x))^alpha.u'(x)` `dx = ((u(x))^(alpha + 1))/(alpha + 1) + c, alpha in RR\\{0,-1}, c in RR` | |
| `int (u'(x))/(u(x))` `dx = ln(abs(u(x))) + c, c in RR` | |
| `int e^u(x).u'(x)` `dx = e^u(x) + c, c in RR` | |
| `int sin(u(x)).u'(x)` `dx = - cos (u(x)) + c, c in RR` | |
| `int cos(u(x)).u'(x)` `dx = sin (u(x)) + c, c in RR` | |
| Linearity rules of integration | `int (f(x) + g(x))` `dx = int f(x)` `dx + int g(x)` `dx` |
| `int k.f(x)` `dx = k int f(x)` `dx` | |
| Integration by parts (or partial integration) | `int u` `dv = uv - int v` `du` |
| Properties of Definite Integrals | `int_b^a f(x)` `dx = - int_a^b f(x)` `dx ` |
| `int_a^a f(x)` `dx = 0` | |
| `int_a^b f(x)` `dx = int_a^c f(x)` `dx + int_c^b f(x)` `dx` | |
| `int_a^b (f(x) + g(x))` `dx = int_a^b f(x)` `dx + int_a^b g(x)` `dx` | |
| `int_a^b k.f(x)` `dx = k int_a^b f(x)` `dx` | |
| Barrow's rule | `int_a^b f(x)` `dx = F(b) - F(a)`, where `F` is primitive from `f` in the interval `[a,b]` |
| Algebraic Form | Complex number | `z = a + bi` | |
| Conjugate | `bar z = a -bi` | ||
| Symmetry | `-z = -a -bi` | ||
| Equality | `a + bi = c + di hArr a = c ^^ b = d` | ||
| Addition | `(a+bi)+(c+di)=(a+c)+(b+d)i` | ||
| Subtraction | `(a+bi)−(c+di)=(a−c)+(b−d)i` | ||
| Multiplication | `(a+bi)xx(c+di)=(ac−bd)+(ad+bc)i` | ||
| Division | `(a+bi)/(c+di)=(a+bi)/(c+di)xx(c−di)/(c−di)=(ac+bd)/(c^2+d^2)+(bc−ad)/(c^2+d^2)i` | ||
| Inverse | `z^-1 = 1/z` | `z^-1 = 1/(|z|^2). bar z` | |
| Properties | `bar bar z = z` | ||
| `|z| = |bar z|` | |||
| `|z|^2 = z.bar z` | |||
| `Re(z) = (z + bar z)/2` | |||
| `Im(z) = (z - bar z)/(2i)` | |||
| Exponential to Algebraic form conversion | Angle | `arg(z) = theta` | `theta = tan^(-1)(b/a)` |
| Distance | `|z|` | `|z| = sqrt(a^2 + b^2)` | |
| Exponential form | Complex number | `z = |z| . e^(i theta)` | `z = |z| . (cos theta + i sin theta)` |
| Conjugate | `bar z = |z| . e^(i(-theta))` | ||
| Symmetry | `-z = |z| . e^(i(theta + pi))` | ||
| Multiplication | `z_1 = |z_1| . e^(i theta_1)` `z_2 = |z_2| . e^(i theta_2)` | `z_1 xx z_2 = |z_1| |z_2| . e^(i (theta_1 + theta_2))` | |
| Division | `z_1 / z_2 = |z_1| / |z_2| . e^(i (theta_1 - theta_2))` | ||
| Exponentiation | `z^n = |z|^n . e^(i n theta)` | ||
| Radicals | `root(n)(|z| . e^(i theta)) = root(n)(|z|) . e^(i ((theta + 2 k pi)/n)), k in {0,...,n-1), n in NN` | ||
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