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The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance.
The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
In mathematics, the definite integral ()is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed ...