Integral of 1/e
Indefinite integral integral 1/e dx = x/e + constant
Derivative of ln(1-x)
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
How to reverse a derivative
Input interpretation string reverse | string: derivative Result evitavired
1/998001
Input 1/998001 Exact result 1/998001 (irreducible) Decimal approximation 1.0020030040050060070080090100110120130140150160170180… × 10^-6 Repeating decimal (period 2997) Prime factorization 3^(-6)×37^(-2)
Integral of sqrt
Indefinite integral integral sqrt(x) dx = (2 x^(3/2))/3 + constant
Derivative of e^-4x
Derivative d/dx(x/e^4) = 1/e^4 Decimal approximation 0.018315638888734180293718021273241242211912067553475594769… Property 1/e^4 is a transcendental number
Integral 1/sqrt(x)
Indefinite integral integral 1/sqrt(x) dx = 2 sqrt(x) + constant
Integral sqrt(1+x^2)
Indefinite integral integral sqrt(1 + x^2) 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…
Integrate cos4x
Indefinite integral integral cos(4 x) dx = 1/4 sin(4 x) + constant Series expansion of the integral at x=0 x – (8 x^3)/3 + (32 x^5)/15 + O(x^6) (Taylor series) Definite integral integral_0^(pi/8) cos(4 x) dx = 1/4 = 0.25 Definite integral mean square integral_0^(pi/2) (2 cos^2(4 x))/pi dx = 1/2 = 0.5
Sqrt(1+x^2) integral
Indefinite integral integral sqrt(1 + x^2) 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…