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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 and and then integrated.
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.
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.
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.
3.1 Integrals of hyperbolic tangent, cotangent, secant, cosecant functions 3.2 Integrals involving hyperbolic sine and cosine functions 3.3 Integrals involving hyperbolic and trigonometric functions
Using the "partitioning the range of f" philosophy, the integral of a non-negative function f : R → R should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f ∗ (t) = μ{ x : f(x) > t}. The Lebesgue integral of f is then defined by
Integrands of the form (A + B x) (a + b x) m (c + d x) n (e + f x) p [ edit ] The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m , n and p toward 0.