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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.
Generally, if the function is any trigonometric function, and is its derivative, ∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C} In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
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. In particular, according to the prime number theorem , it is a very good approximation to the prime-counting function , which is defined as the number of prime numbers ...
Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, Si(x) is the antiderivative of sin x / x whose value is zero at x = 0, and si(x) is the antiderivative whose value is zero at x = ∞.
Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation): a = e ln a , {\displaystyle a=e^{\ln a},} and that e a e b = e a + b , {\displaystyle e^{a}e^{b}=e^{a+b},} both valid for any complex ...
The tangent half-angle substitution relates an angle to the slope of a line. Introducing a new variable = , sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows: = +, = +, = +.
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.
CORDIC (coordinate rotation digital computer), Volder's algorithm, Digit-by-digit method, Circular CORDIC (Jack E. Volder), [1] [2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), [3] [4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), [5] [6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots ...