List of integrals of trigonometric 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. 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,

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

Integrands involving only sine

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Integrands involving only cosine

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Integrands involving only tangent

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Integrands involving only secant

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Integrands involving only cosecant

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Integrands involving only cotangent

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Integrands involving both sine and cosine

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An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.

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Integrands involving both sine and tangent

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Integrand involving both cosine and tangent

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Integrand involving both sine and cotangent

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Integrand involving both cosine and cotangent

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Integrand involving both secant and tangent

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Integrand involving both cosecant and cotangent

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Integrals in a quarter period

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Using the beta function   one can write

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Integrals with symmetric limits

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Integral over a full circle

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See also

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References

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  1. ^ Bresock, Krista, "Student Understanding of the Definite Integral When Solving Calculus Volume Problems" (2022). Graduate Theses, Dissertations, and Problem Reports. 11491. https://researchrepository.wvu.edu/etd/11491