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. Generally, if the function sin x {\displaystyle \sin x} is any trigonometric function, and cos x {\displaystyle \cos x} is its derivative,
∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C}
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
Integrands involving only sine
Integrands involving only cosine
Integrands involving only tangent
Integrands involving only secant
Integrands involving only cosecant
Integrands involving only cotangent
Integrands involving both sine and cosine 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
Integrand involving both cosine and tangent
Integrand involving both sine and cotangent
Integrand involving both cosine and cotangent
Integrand involving both secant and tangent
Integrand involving both cosecant and cotangent
Integrals in a quarter period Using the beta function B ( a , b ) {\displaystyle B(a,b)} one can write
Using the modified Struve functions L α ( x ) {\displaystyle L_{\alpha }(x)} and modified Bessel functions I α ( x ) {\displaystyle I_{\alpha }(x)} one can write
Integrals with symmetric limits
Integral over a full circle
See also Trigonometric integral
References