Indefinite integral integral 1/(sin(x)) dx = -log(cot(x) + csc(x)) + constant Series expansion of the integral at x=0 log(x/2) + x^2/12 + (7 x^4)/1440 + O(x^6) (generalized Puiseux series) Series expansion of the integral at x=-π (-log(x + pi) + i pi + log(2)) – 1/12 (x + pi)^2 – (7 (x + pi)^4)/1440 + O((x + pi)^6) (generalized Puiseux series) Series expansion of the integral at x=π (-log(x – pi) – i pi + log(2)) – 1/12 (x –…
Derivative d/dx(e^4 x) = e^4 Decimal approximation 54.59815003314423907811026120286087840279073703861406872582… Property e^4 is a transcendental number
Indefinite integral integral 1/sqrt(x) dx = 2 sqrt(x) + constant
Input interpretation isosceles right triangle | area Result A = a^2/2 Defining inequalities y>=0 and a>=x + y and x>=0
Input interpretation prime number | detailed definition Result A prime number (or prime integer, often simply called a “prime” for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime…
Input 2 sqrt(x) Root x = 0 Derivative d/dx(2 sqrt(x)) = 1/sqrt(x) Indefinite integral integral 2 sqrt(x) dx = (4 x^(3/2))/3 + constant Global minimum min{2 sqrt(x)} = 0 at x = 0
Input sqrt(1 + x^2) Series expansion at x=0 1 + x^2/2 – x^4/8 + O(x^5) (Taylor series) Series expansion at x=-i sqrt(2 – 2 i x) + (i sqrt(1 – i x) (x + i))/(2 sqrt(2)) + (sqrt(1 – i x) (x + i)^2)/(16 sqrt(2)) – (i sqrt(1 – i x) (x + i)^3)/(64 sqrt(2)) – (5 sqrt(1 – i x) (x + i)^4)/(1024 sqrt(2)) + O((x + i)^5) (generalized Puiseux series) Series expansion at x=i sqrt(2 + 2 i…
Indefinite integral integral x sin(x) dx = sin(x) – x cos(x) + constant Alternate form of the integral -1/2 e^(-i x) x – 1/2 e^(i x) x + 1/2 i e^(-i x) – 1/2 i e^(i x) + constant Series expansion of the integral at x=0 x^3/3 – x^5/30 + x^7/840 + O(x^8) (Taylor series) Definite integral integral_0^pi x sin(x) dx = pi~~3.14159
Indefinite integral integral sqrt(x) dx = (2 x^(3/2))/3 + constant
Indefinite integral integral sqrt(x) dx = (2 x^(3/2))/3 + constant