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For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration . Integrals involving only hyperbolic sine functions
The following is a list of integrals (antiderivative functions) of trigonometric functions.For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
The following Python code with the SymPy library will allow for calculation of the values of and to 20 digits of precision: from sympy import * def lag_weights_roots ( n ): x = Symbol ( "x" ) roots = Poly ( laguerre ( n , x )) . all_roots () x_i = [ rt . evalf ( 20 ) for rt in roots ] w_i = [( rt / (( n + 1 ) * laguerre ( n + 1 , rt )) ** 2 ...
to transform an integral on the interval x ∈ (−1, 1) to an integral on the entire real line t ∈ (−∞, ∞), the two integrals having the same value. After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula .
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 ...
At this point we can either integrate directly, or we can first change the integrand to 2 cos 6x − 4 cos 4x + 2 cos 2x and continue from there. Either method gives Either method gives ∫ sin 2 x cos 4 x d x = − 1 24 sin 6 x + 1 8 sin 4 x − 1 8 sin 2 x + C . {\displaystyle \int \sin ^{2}x\cos 4x\,dx=-{\frac {1}{24 ...
The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut. The arcsine function may then be defined as: