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These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
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. For a complete list of antiderivative functions, see Lists of integrals.
Because integration above requires that / < < /, can only go from to / Neglecting this restriction, one might have picked θ {\displaystyle \theta } to go from π {\displaystyle \pi } to 5 π / 6 , {\displaystyle 5\pi /6,} which would have resulted in the negative of the actual value.
Finally, L'Hôpital's rule tells us that = =, which rearranges to for small values of θ. Alternatively, we can use the double angle formula cos 2 A ≡ 1 − 2 sin 2 A {\displaystyle \cos 2A\equiv 1-2\sin ^{2}A} .
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
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.
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.
Bioche's rules, formulated by the French mathematician Charles Bioche (1859–1949), are rules to aid in the computation of certain indefinite integrals in which the integrand contains sines and cosines. In the following, () is a rational expression in and .