List of integrals of trigonometric functions

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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][2]

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[edit]

Integrands involving only cosine[edit]

Integrands involving only tangent[edit]

Integrands involving only secant[edit]

Integrands involving only cosecant[edit]

Integrands involving only cotangent[edit]

Integrands involving both sine and cosine[edit]

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

Integrands involving both sine and tangent[edit]

Integrand involving both cosine and tangent[edit]

Integrand involving both sine and cotangent[edit]

Integrand involving both cosine and cotangent[edit]

Integrand involving both secant and tangent[edit]

Integrand involving both cosecant and cotangent[edit]

Integrals in a quarter period[edit]

Using the beta function one can write

Integrals with symmetric limits[edit]

Integral over a full circle[edit]

See also[edit]

References[edit]

  1. ^ {{cite news}}: Empty citation (help)
  2. ^ 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