HOME

TheInfoList



OR:

The following is a list of
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s (
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
functions) of
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
. 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 is any trigonometric function, and \cos x is its derivative, : \int a\cos nx\,dx = \frac\sin nx+C In all formulas the constant ''a'' is assumed to be nonzero, and ''C'' denotes the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
.


Integrands involving only

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...

: \int\sin ax\,dx = -\frac\cos ax+C : \int\sin^2 \,dx = \frac - \frac \sin 2ax +C= \frac - \frac \sin ax\cos ax +C : \int\sin^3 \,dx = \frac - \frac +C : \int x\sin^2 \,dx = \frac - \frac \sin 2ax - \frac \cos 2ax +C : \int x^2\sin^2 \,dx = \frac - \left( \frac - \frac \right) \sin 2ax - \frac \cos 2ax +C :\int x\sin ax\,dx = \frac-\frac+C : \int(\sin b_1x)(\sin b_2x)\,dx = \frac-\frac+C \qquad\mbox, b_1, \neq, b_2, \mbox : \int\sin^n \,dx = -\frac + \frac\int\sin^ ax\,dx \qquad\mboxn>0\mbox : \int\frac = -\frac\ln+C : \int\frac = \frac+\frac\int\frac \qquad\mboxn>1\mbox : \begin \int x^n\sin ax\,dx &= -\frac\cos ax+\frac\int x^\cos ax\,dx \\ &= \sum_^ (-1)^ \frac\frac \cos ax +\sum_^(-1)^k \frac\frac \sin ax \\ &= - \sum_^n \frac\frac\cos\left(ax+k\frac\right) \qquad\mboxn>0\mbox \end : : \int\frac\,dx = \sum_^\infty (-1)^n\frac +C : \int\frac\,dx = -\frac + \frac\int\frac\,dx : \int\mathrm\left\}\cos\left(\right)\left(\right)\mathrm\sqrt\sqrt\sin\left(\right)\left(\right)\;\;^\mathrm\;\;}\\ & \end}\right.\;\;\;\diagup\!\!\!\!\; : \int\frac = \frac\tan\left(\frac\mp\frac\right)+C : \int\frac = \frac\tan\left(\frac - \frac\right)+\frac\ln\left, \cos\left(\frac-\frac\right)\+C : \int\frac = \frac\cot\left(\frac - \frac\right)+\frac\ln\left, \sin\left(\frac-\frac\right)\+C : \int\frac = \pm x+\frac\tan\left(\frac\mp\frac\right)+C


Integrands involving only cosine

: \int\cos ax\,dx = \frac\sin ax+C : \int\cos^2 \,dx = \frac + \frac \sin 2ax +C = \frac + \frac \sin ax\cos ax +C : \int\cos^n ax\,dx = \frac + \frac\int\cos^ ax\,dx \qquad\mboxn>0\mbox : \int x\cos ax\,dx = \frac + \frac+C : \int x^2\cos^2 \,dx = \frac + \left( \frac - \frac \right) \sin 2ax + \frac \cos 2ax +C : \begin \int x^n\cos ax\,dx &= \frac - \frac\int x^\sin ax\,dx \\ &= \sum_^ (-1)^ \frac\frac \cos ax +\sum_^(-1)^ \frac\frac \sin ax \\ &=\sum_^n (-1)^ \frac\frac\cos\left(ax -\frac\frac\right) \\ &=\sum_^n \frac\frac\sin\left(ax+k\frac\right) \qquad\mboxn>0\mbox \end : \int\frac\,dx = \ln, ax, +\sum_^\infty (-1)^k\frac+C : \int\frac\,dx = -\frac-\frac\int\frac\,dx \qquad\mboxn\neq 1\mbox : \int\frac = \frac\ln\left, \tan\left(\frac+\frac\right)\+C : \int\frac = \frac + \frac\int\frac \qquad\mboxn>1\mbox : \int\frac = \frac\tan\frac+C : \int\frac = -\frac\cot\frac+C : \int\frac = \frac\tan\frac + \frac\ln\left, \cos\frac\+C : \int\frac = -\frac\cot\frac+\frac\ln\left, \sin\frac\+C : \int\frac = x - \frac\tan\frac+C : \int\frac = -x-\frac\cot\frac+C : \int(\cos a_1x)(\cos a_2x)\,dx = \frac+\frac+C \qquad\mbox, a_1, \neq, a_2, \mbox


Integrands involving only

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

: \int\tan ax\,dx = -\frac\ln, \cos ax, +C = \frac\ln, \sec ax, +C :\int \tan^2 \, dx = \tan - x +C : \int\tan^n ax\,dx = \frac\tan^ ax-\int\tan^ ax\,dx \qquad\mboxn\neq 1\mbox : \int\frac = \frac(px + \frac\ln, q\sin ax + p\cos ax, )+C \qquad\mboxp^2 + q^2\neq 0\mbox : \int\frac = \pm \frac + \frac\ln, \sin ax \pm \cos ax, +C : \int\frac = \frac \mp \frac\ln, \sin ax \pm \cos ax, +C


Integrands involving only secant

: ''See
Integral of the secant function In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities, : \int \sec \thet ...
.'' :\int \sec \, dx = \frac\ln+C= \frac\ln+C = \frac\operatorname+C :\int \sec^2 \, dx = \tan+C :\int \sec^3 \, dx = \frac\sec x \tan x + \frac\ln, \sec x + \tan x, + C. :\int \sec^n \, dx = \frac \,+\, \frac\int \sec^ \, dx \qquad \mboxn \ne 1\mbox :\int \frac = x - \tan+C :\int \frac = - x - \cot+C


Integrands involving only

cosecant In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

:\int \csc \, dx= -\frac\ln+C= \frac\ln+C = \frac\ln+C :\int \csc^2 \, dx = -\cot+C :\int \csc^3 \, dx = -\frac\csc x \cot x - \frac\ln, \csc x + \cot x, + C = -\frac\csc x \cot x + \frac\ln, \csc x - \cot x, + C :\int \csc^n \, dx = -\frac \,+\, \frac\int \csc^ \, dx \qquad \mboxn \ne 1\mbox :\int \frac = x - \frac+C :\int \frac = - x + \frac+C


Integrands involving only

cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

:\int\cot ax\,dx = \frac\ln, \sin ax, +C :\int \cot^2 \, dx = -\cot - x +C : \int\cot^n ax\,dx = -\frac\cot^ ax - \int\cot^ ax\,dx \qquad\mboxn\neq 1\mbox : \int\frac = \int\frac = \frac - \frac\ln, \sin ax + \cos ax, +C : \int\frac = \int\frac = \frac + \frac\ln, \sin ax - \cos ax, +C


Integrands involving both

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and cosine

An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules. : \int\frac = \frac\ln\left, \tan\left(\frac\pm\frac\right)\+C : \int\frac = \frac\tan\left(ax\mp\frac\right)+C : \int\frac = \frac\left(\frac + (n - 2)\int\frac \right) : \int\frac = \frac + \frac\ln\left, \sin ax + \cos ax\+C : \int\frac = \frac - \frac\ln\left, \sin ax - \cos ax\+C : \int\frac = \frac - \frac\ln\left, \sin ax + \cos ax\+C : \int\frac = -\frac - \frac\ln\left, \sin ax - \cos ax\+C : \int\frac = -\frac\tan^2\frac+\frac\ln\left, \tan\frac\+C : \int\frac = -\frac\cot^2\frac-\frac\ln\left, \tan\frac\+C : \int\frac = \frac\cot^2\left(\frac+\frac\right)+\frac\ln\left, \tan\left(\frac+\frac\right)\+C : \int\frac = \frac\tan^2\left(\frac+\frac\right)-\frac\ln\left, \tan\left(\frac+\frac\right)\+C : \int(\sin ax)(\cos ax)\,dx = \frac\sin^2 ax +C : \int(\sin a_1x)(\cos a_2x)\,dx = -\frac -\frac +C\qquad\mbox, a_1, \neq, a_2, \mbox : \int(\sin^n ax)(\cos ax)\,dx = \frac\sin^ ax +C\qquad\mboxn\neq -1\mbox : \int(\sin ax)(\cos^n ax)\,dx = -\frac\cos^ ax +C\qquad\mboxn\neq -1\mbox : \begin \int(\sin^n ax)(\cos^m ax)\,dx &= -\frac+\frac\int(\sin^ ax)(\cos^m ax)\,dx \qquad\mboxm,n>0\mbox \\ &= \frac + \frac\int(\sin^n ax)(\cos^ ax)\,dx \qquad\mboxm,n>0\mbox \end : \int\frac = \frac\ln\left, \tan ax\+C : \int\frac = \frac+\int\frac \qquad\mboxn\neq 1\mbox : \int\frac = -\frac+\int\frac \qquad\mboxn\neq 1\mbox : \int\frac = \frac +C\qquad\mboxn\neq 1\mbox : \int\frac = -\frac\sin ax+\frac\ln\left, \tan\left(\frac+\frac\right)\+C : \int\frac = \frac-\frac\int\frac \qquad\mboxn\neq 1\mbox : \begin \int \frac \, dx &= \sqrt\operatorname\left(\frac\right) - x \qquad\mbox] - \frac ; + \frac [\mbox \\ &= \sqrt\operatorname\left(\frac\right)-\operatorname\left(\tan x\right) \qquad\mbox\mbox \end : \int\frac = -\frac + \int\frac \qquad\mboxn\neq 1\mbox : \int\frac = \begin \frac-\frac\int\frac &\mboxm\neq 1\mbox \\ \frac-\frac\int\frac &\mboxm\neq 1\mbox \\ -\frac+\frac\int\frac &\mboxm\neq n\mbox \end : \int\frac = -\frac +C\qquad\mboxn\neq 1\mbox : \int\frac = \frac\left(\cos ax+\ln\left, \tan\frac\\right) +C : \int\frac = -\frac\left(\frac+\int\frac\right) \qquad\mboxn\neq 1\mbox : \int\frac = \begin -\frac - \frac\int\frac &\mboxm\neq 1\mbox \\ -\frac - \frac\int\frac &\mboxm\neq 1\mbox \\ \frac + \frac\int\frac &\mboxm\neq n\mbox \end


Integrands involving both

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

: \int (\sin ax)(\tan ax)\,dx = \frac(\ln, \sec ax + \tan ax, - \sin ax)+C : \int\frac = \frac\tan^ (ax) +C\qquad\mboxn\neq 1\mbox


Integrand involving both cosine and

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

: \int\frac = \frac\tan^ ax +C\qquad\mboxn\neq -1\mbox


Integrand involving both

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and
cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

: \int\frac = -\frac\cot^ ax +C\qquad\mboxn\neq -1\mbox


Integrand involving both cosine and

cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

: \int\frac = \frac\tan^ ax +C\qquad\mboxn\neq 1\mbox


Integrand involving both secant and

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

: \int(\sec x)(\tan x)\,dx= \sec x + C


Integrand involving both

cosecant In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
and
cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

: \int(\csc x)(\cot x)\,dx= -\csc x + C


Integrals in a quarter period

: \int_^} \sin^n x \, dx = \int_^} \cos^n x \, dx = \begin \frac \cdot \frac \cdots \frac \cdot \frac \cdot \frac, & \text n\text \\ \frac \cdot \frac \cdots \frac \cdot \frac, & \text n\text \\ 1, & \text n=1 \end


Integrals with symmetric limits

: \int_^\sin\,dx = 0 : \int_^\cos \,dx = 2\int_^\cos \,dx = 2\int_^\cos \,dx = 2\sin : \int_^\tan \,dx = 0 : \int_^ x^2\cos^2 \,dx = \frac \qquad\mboxn=1,3,5...\mbox : \int_^ x^2\sin^2 \,dx = \frac = \frac (1-6\frac) \qquad\mboxn=1,2,3,...\mbox


Integral over a full circle

: \int_^\sin^\cos^\,dx = 0 \! \qquad n,m \in \mathbb : \int_^\sin^\cos^\,dx = 0 \! \qquad n,m \in \mathbb


See also

* Trigonometric integral {{DEFAULTSORT:Integrals of Trigonometric Functions
Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
Trigonometry