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The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity 1 + cot 2 θ = csc 2 θ {\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta } , and the red triangle shows that tan 2 θ + 1 = sec 2 θ {\displaystyle \tan ^{2 ...
For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral. [1] Generally, if the function is any trigonometric function, and is its derivative,
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: = +, = +, = +.
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.
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [ 2 ]
Si(x) (blue) and Ci(x) (green) shown on the same plot. Sine integral in the complex plane, plotted with a variant of domain coloring. Cosine integral in the complex plane. Note the branch cut along the negative real axis. In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
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.
For a complete list of integral formulas, see lists of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration. For each inverse hyperbolic integration formula below there is a corresponding formula in the list of integrals of inverse trigonometric functions.