Indefinite integral integral sqrt(1 – x^2) dx = 1/2 (sqrt(1 – x^2) x + sin^(-1)(x)) + constant Expanded form of the integral 1/2 sqrt(1 – x^2) x + 1/2 sin^(-1)(x) + constant Series expansion of the integral at x=-1 -pi/4 + 2/3 sqrt(2) (x + 1)^(3/2) – (x + 1)^(5/2)/(5 sqrt(2)) + O((x + 1)^(7/2)) (Puiseux series) Series expansion of the integral at x=0 x – x^3/6 – x^5/40 – x^7/112 + O(x^9) (Taylor series) Series expansion of the integral…
Derivative d/dx(log(1/x)) = -1/x Roots (no roots exist) Indefinite integral integral -1/x dx = -log(x) + constant Limit lim_(x-> ± infinity) -1/x = 0
Derivative d/dx(log(1) x) = 0 Number name zero
Indefinite integral integral 1/sqrt(x) dx = 2 sqrt(x) + constant
Input sqrt(1 – x^2) Alternate form sqrt(1 – x) sqrt(x + 1) Series expansion at x=-1 sqrt(2) sqrt(x + 1) – (x + 1)^(3/2)/(2 sqrt(2)) – (x + 1)^(5/2)/(16 sqrt(2)) + O((x + 1)^(7/2)) (Puiseux series) Series expansion at x=0 1 – x^2/2 – x^4/8 + O(x^5) (Taylor series) Series expansion at x=1 i sqrt(2) sqrt(x – 1) + (i (x – 1)^(3/2))/(2 sqrt(2)) – (i (x – 1)^(5/2))/(16 sqrt(2)) + O((x – 1)^(7/2)) (Puiseux series) Series expansion at x=∞…
Input interpretation Joe Namath (football player) Basic information full name | Joseph William Namath date of birth | Monday, May 31, 1943 (age: 74 years) place of birth | Beaver Falls, Pennsylvania, United States NFL player information team history | New York Jets (1965 to 1976) | Los Angeles Rams (1977) position | quarterback (data since 1985-86 season) Physical characteristics height | 1.88 meters weight | 91 kg (kilograms) Notable films Norwood (1970)
Derivative d/dx(log(1 + x)) = 1/(x + 1) Roots (no roots exist) Series expansion at x=0 1 – x + x^2 – x^3 + x^4 + O(x^5) (Taylor series) Series expansion at x=∞ 1/x – (1/x)^2 + (1/x)^3 – (1/x)^4 + O((1/x)^5) (Laurent series) Indefinite integral integral 1/(1 + x) dx = log(x + 1) + constant Limit lim_(x-> ± infinity) 1/(1 + x) = 0
Input interpretation series | sqrt(1 + x) Series expansion at x=0 1 + x/2 – x^2/8 + x^3/16 – (5 x^4)/128 + (7 x^5)/256 + O(x^6) (Taylor series) (converges when abs(x)
Indefinite integral integral sqrt(x^2 + 1) dx = 1/2 (sqrt(x^2 + 1) x + sinh^(-1)(x)) + constant Alternate form of the integral 1/2 sqrt(x^2 + 1) x + 1/2 log(sqrt(x^2 + 1) + x) + constant Expanded form of the integral 1/2 sqrt(x^2 + 1) x + 1/2 sinh^(-1)(x) + constant Series expansion of the integral at x=0 x + x^3/6 – x^5/40 + x^7/112 + O(x^9) (Taylor series) Series expansion of the integral at x=-i (-1)^(floor(((3 pi)/2 – arg(x…
Indefinite integral integral sqrt(1 – x^2) dx = 1/2 (sqrt(1 – x^2) x + sin^(-1)(x)) + constant Expanded form of the integral 1/2 sqrt(1 – x^2) x + 1/2 sin^(-1)(x) + constant Definite integral integral_(-1)^1 sqrt(1 – x^2) dx = pi/2~~1.5708