List of integrals of trigonometric functions


List of integrals of trigonometric functions

The following is a list of integrals (antiderivative functions) of trigonometric functions. For integrals involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of Integral functions, see lists of integrals. See also trigonometric integral.

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

Integrals containing only sine

: intsin ax;dx = -frac{1}{a}cos ax+C,!

: intsin^2 {ax};dx = frac{x}{2} - frac{1}{4a} sin 2ax +C= frac{x}{2} - frac{1}{2a} sin axcos ax +C!

: int xsin^2 {ax};dx = frac{x^2}{4} - frac{x}{4a} sin 2ax - frac{1}{8a^2} cos 2ax +C!

: int x^2sin^2 {ax};dx = frac{x^3}{6} - left( frac {x^2}{4a} - frac{1}{8a^3} ight) sin 2ax - frac{x}{4a^2} cos 2ax +C!

: intsin b_1xsin b_2x;dx = frac{sin [(b_1-b_2)x] }{2(b_1-b_2)}-frac{sin [(b_1+b_2)x] }{2(b_1+b_2)}+C qquadmbox{(for }|b_1| eq|b_2|mbox{)},!

: intsin^n {ax};dx = -frac{sin^{n-1} axcos ax}{na} + frac{n-1}{n}intsin^{n-2} ax;dx qquadmbox{(for }n>0mbox{)},!

: intfrac{dx}{sin ax} = frac{1}{a}ln left| anfrac{ax}{2} ight|+C

: intfrac{dx}{sin^n ax} = frac{cos ax}{a(1-n) sin^{n-1} ax}+frac{n-2}{n-1}intfrac{dx}{sin^{n-2}ax} qquadmbox{(for }n>1mbox{)},!

: int xsin ax;dx = frac{sin ax}{a^2}-frac{xcos ax}{a}+C,!

: int x^nsin ax;dx = -frac{x^n}{a}cos ax+frac{n}{a}int x^{n-1}cos ax;dx qquadmbox{(for }n>0mbox{)},!

: int_{frac{-a}{2^{frac{a}{2 x^2sin^2 {frac{npi x}{a;dx = frac{a^3(n^2pi^2-6)}{24n^2pi^2} qquadmbox{(for }n=2,4,6...mbox{)},!

: intfrac{sin ax}{x} dx = sum_{n=0}^infty (-1)^nfrac{(ax)^{2n+1{(2n+1)cdot (2n+1)!} +C,!

: intfrac{sin ax}{x^n} dx = -frac{sin ax}{(n-1)x^{n-1 + frac{a}{n-1}intfrac{cos ax}{x^{n-1 dx,!

: intfrac{dx}{1pmsin ax} = frac{1}{a} anleft(frac{ax}{2}mpfrac{pi}{4} ight)+C

: intfrac{x;dx}{1+sin ax} = frac{x}{a} anleft(frac{ax}{2} - frac{pi}{4} ight)+frac{2}{a^2}lnleft|cosleft(frac{ax}{2}-frac{pi}{4} ight) ight|+C

: intfrac{x;dx}{1-sin ax} = frac{x}{a}cotleft(frac{pi}{4} - frac{ax}{2} ight)+frac{2}{a^2}lnleft|sinleft(frac{pi}{4}-frac{ax}{2} ight) ight|+C

: intfrac{sin ax;dx}{1pmsin ax} = pm x+frac{1}{a} anleft(frac{pi}{4}mpfrac{ax}{2} ight)+C

Integrals containing only cosine

: intcos ax;dx = frac{1}{a}sin ax+C,!

: intcos^n ax;dx = frac{cos^{n-1} axsin ax}{na} + frac{n-1}{n}intcos^{n-2} ax;dx qquadmbox{(for }n>0mbox{)},!

: int xcos ax;dx = frac{cos ax}{a^2} + frac{xsin ax}{a}+C,!

: intcos^2 {ax};dx = frac{x}{2} + frac{1}{4a} sin 2ax +C = frac{x}{2} + frac{1}{2a} sin axcos ax +C!

: int x^2cos^2 {ax};dx = frac{x^3}{6} + left( frac {x^2}{4a} - frac{1}{8a^3} ight) sin 2ax + frac{x}{4a^2} cos 2ax +C!

: int x^ncos ax;dx = frac{x^nsin ax}{a} - frac{n}{a}int x^{n-1}sin ax;dx,!

: int_{frac{-a}{2^{frac{a}{2 x^2cos^2 {frac{npi x}{a;dx = frac{a^3(n^2pi^2-6)}{24n^2pi^2} qquadmbox{(for }n=1,3,5...mbox{)},!

: intfrac{cos ax}{x} dx = ln|ax|+sum_{k=1}^infty (-1)^kfrac{(ax)^{2k{2kcdot(2k)!}+C,!

: intfrac{cos ax}{x^n} dx = -frac{cos ax}{(n-1)x^{n-1-frac{a}{n-1}intfrac{sin ax}{x^{n-1 dx qquadmbox{(for }n eq 1mbox{)},!

: intfrac{dx}{cos ax} = frac{1}{a}lnleft| anleft(frac{ax}{2}+frac{pi}{4} ight) ight|+C

: intfrac{dx}{cos^n ax} = frac{sin ax}{a(n-1) cos^{n-1} ax} + frac{n-2}{n-1}intfrac{dx}{cos^{n-2} ax} qquadmbox{(for }n>1mbox{)},!

: intfrac{dx}{1+cos ax} = frac{1}{a} anfrac{ax}{2}+C,!

: intfrac{dx}{1-cos ax} = -frac{1}{a}cotfrac{ax}{2}+C,!

: intfrac{x;dx}{1+cos ax} = frac{x}{a} anfrac{ax}{2} + frac{2}{a^2}lnleft|cosfrac{ax}{2} ight|+C

: intfrac{x;dx}{1-cos ax} = -frac{x}{a}cotfrac{ax}{2}+frac{2}{a^2}lnleft|sinfrac{ax}{2} ight|+C

: intfrac{cos ax;dx}{1+cos ax} = x - frac{1}{a} anfrac{ax}{2}+C,!

: intfrac{cos ax;dx}{1-cos ax} = -x-frac{1}{a}cotfrac{ax}{2}+C,!

: intcos a_1xcos a_2x;dx = frac{sin(a_1-a_2)x}{2(a_1-a_2)}+frac{sin(a_1+a_2)x}{2(a_1+a_2)}+C qquadmbox{(for }|a_1| eq|a_2|mbox{)},!

Integrals containing only tangent

: int an ax;dx = -frac{1}{a}ln|cos ax|+C = frac{1}{a}ln|sec ax|+C,!

: int an^n ax;dx = frac{1}{a(n-1)} an^{n-1} ax-int an^{n-2} ax;dx qquadmbox{(for }n eq 1mbox{)},!

: intfrac{dx}{q an ax + p} = frac{1}{p^2 + q^2}(px + frac{q}{a}ln|qsin ax + pcos ax|)+C qquadmbox{(for }p^2 + q^2 eq 0mbox{)},!

: intfrac{dx}{ an ax} = frac{1}{a}ln|sin ax|+C,!

: intfrac{dx}{ an ax + 1} = frac{x}{2} + frac{1}{2a}ln|sin ax + cos ax|+C,!

: intfrac{dx}{ an ax - 1} = -frac{x}{2} + frac{1}{2a}ln|sin ax - cos ax|+C,!

: intfrac{ an ax;dx}{ an ax + 1} = frac{x}{2} - frac{1}{2a}ln|sin ax + cos ax|+C,!

: intfrac{ an ax;dx}{ an ax - 1} = frac{x}{2} + frac{1}{2a}ln|sin ax - cos ax|+C,!

Integrals containing only secant

:int sec{ax} , dx = frac{1}{a}ln{left| sec{ax} + an{ax} ight+C

:int sec^n{ax} , dx = frac{sec^{n-1}{ax} sin {ax{a(n-1)} ,+, frac{n-2}{n-1}int sec^{n-2}{ax} , dx qquad mbox{ (for }n e 1mbox{)},!

:int sec^n{x} , dx = frac{sec^{n-2}{x} an{x{n-1} ,+, frac{n-2}{n-1}int sec^{n-2}{x},dx [Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008]

:int frac{dx}{sec{x} + 1} = x - an{frac{x}{2+C

Integrals containing only cosecant

:int csc{ax} , dx = -frac{1}{a}ln{left| csc{ax} + cot{ax} ight+C

:int csc^2{x} , dx = -cot{x}+C

:int csc^n{ax} , dx = -frac{csc^{n-1}{ax} cos{ax{a(n-1)} ,+, frac{n-2}{n-1}int csc^{n-2}{ax} , dx qquad mbox{ (for }n e 1mbox{)},!

Integrals containing only cotangent

:intcot ax;dx = frac{1}{a}ln|sin ax|+C,!

: intcot^n ax;dx = -frac{1}{a(n-1)}cot^{n-1} ax - intcot^{n-2} ax;dx qquadmbox{(for }n eq 1mbox{)},!

: intfrac{dx}{1 + cot ax} = intfrac{ an ax;dx}{ an ax+1},!

: intfrac{dx}{1 - cot ax} = intfrac{ an ax;dx}{ an ax-1},!

Integrals containing both sine and cosine

: intfrac{dx}{cos axpmsin ax} = frac{1}{asqrt{2lnleft| anleft(frac{ax}{2}pmfrac{pi}{8} ight) ight|+C

: intfrac{dx}{(cos axpmsin ax)^2} = frac{1}{2a} anleft(axmpfrac{pi}{4} ight)+C

: intfrac{dx}{(cos x + sin x)^n} = frac{1}{n-1}left(frac{sin x - cos x}{(cos x + sin x)^{n - 1 - 2(n - 2)intfrac{dx}{(cos x + sin x)^{n-2 ight)

: intfrac{cos ax;dx}{cos ax + sin ax} = frac{x}{2} + frac{1}{2a}lnleft|sin ax + cos ax ight|+C

: intfrac{cos ax;dx}{cos ax - sin ax} = frac{x}{2} - frac{1}{2a}lnleft|sin ax - cos ax ight|+C

: intfrac{sin ax;dx}{cos ax + sin ax} = frac{x}{2} - frac{1}{2a}lnleft|sin ax + cos ax ight|+C

: intfrac{sin ax;dx}{cos ax - sin ax} = -frac{x}{2} - frac{1}{2a}lnleft|sin ax - cos ax ight|+C

: intfrac{cos ax;dx}{sin ax(1+cos ax)} = -frac{1}{4a} an^2frac{ax}{2}+frac{1}{2a}lnleft| anfrac{ax}{2} ight|+C

: intfrac{cos ax;dx}{sin ax(1+-cos ax)} = -frac{1}{4a}cot^2frac{ax}{2}-frac{1}{2a}lnleft| anfrac{ax}{2} ight|+C

: intfrac{sin ax;dx}{cos ax(1+sin ax)} = frac{1}{4a}cot^2left(frac{ax}{2}+frac{pi}{4} ight)+frac{1}{2a}lnleft| anleft(frac{ax}{2}+frac{pi}{4} ight) ight|+C

: intfrac{sin ax;dx}{cos ax(1-sin ax)} = frac{1}{4a} an^2left(frac{ax}{2}+frac{pi}{4} ight)-frac{1}{2a}lnleft| anleft(frac{ax}{2}+frac{pi}{4} ight) ight|+C

: intsin axcos ax;dx = frac{1}{2a}sin^2 ax +c,!

: intsin a_1xcos a_2x;dx = -frac{cos(a_1+a_2)x}{2(a_1+a_2)}-frac{cos(a_1-a_2)x}{2(a_1-a_2)} +Cqquadmbox{(for }|a_1| eq|a_2|mbox{)},!

: intsin^n axcos ax;dx = frac{1}{a(n+1)}sin^{n+1} ax +Cqquadmbox{(for }n eq -1mbox{)},!

: intsin axcos^n ax;dx = -frac{1}{a(n+1)}cos^{n+1} ax +Cqquadmbox{(for }n eq -1mbox{)},!

: intsin^n axcos^m ax;dx = -frac{sin^{n-1} axcos^{m+1} ax}{a(n+m)}+frac{n-1}{n+m}intsin^{n-2} axcos^m ax;dx qquadmbox{(for }m,n>0mbox{)},!

: also: intsin^n axcos^m ax;dx = frac{sin^{n+1} axcos^{m-1} ax}{a(n+m)} + frac{m-1}{n+m}intsin^n axcos^{m-2} ax;dx qquadmbox{(for }m,n>0mbox{)},!

: intfrac{dx}{sin axcos ax} = frac{1}{a}lnleft| an ax ight|+C

: intfrac{dx}{sin axcos^n ax} = frac{1}{a(n-1)cos^{n-1} ax}+intfrac{dx}{sin axcos^{n-2} ax} qquadmbox{(for }n eq 1mbox{)},!

: intfrac{dx}{sin^n axcos ax} = -frac{1}{a(n-1)sin^{n-1} ax}+intfrac{dx}{sin^{n-2} axcos ax} qquadmbox{(for }n eq 1mbox{)},!

: intfrac{sin ax;dx}{cos^n ax} = frac{1}{a(n-1)cos^{n-1} ax} +Cqquadmbox{(for }n eq 1mbox{)},!

: intfrac{sin^2 ax;dx}{cos ax} = -frac{1}{a}sin ax+frac{1}{a}lnleft| anleft(frac{pi}{4}+frac{ax}{2} ight) ight|+C

: intfrac{sin^2 ax;dx}{cos^n ax} = frac{sin ax}{a(n-1)cos^{n-1}ax}-frac{1}{n-1}intfrac{dx}{cos^{n-2}ax} qquadmbox{(for }n eq 1mbox{)},!

: intfrac{sin^n ax;dx}{cos ax} = -frac{sin^{n-1} ax}{a(n-1)} + intfrac{sin^{n-2} ax;dx}{cos ax} qquadmbox{(for }n eq 1mbox{)},!

: intfrac{sin^n ax;dx}{cos^m ax} = frac{sin^{n+1} ax}{a(m-1)cos^{m-1} ax}-frac{n-m+2}{m-1}intfrac{sin^n ax;dx}{cos^{m-2} ax} qquadmbox{(for }m eq 1mbox{)},!

: also: intfrac{sin^n ax;dx}{cos^m ax} = -frac{sin^{n-1} ax}{a(n-m)cos^{m-1} ax}+frac{n-1}{n-m}intfrac{sin^{n-2} ax;dx}{cos^m ax} qquadmbox{(for }m eq nmbox{)},!

: also: intfrac{sin^n ax;dx}{cos^m ax} = frac{sin^{n-1} ax}{a(m-1)cos^{m-1} ax}-frac{n-1}{m-1}intfrac{sin^{n-2} ax;dx}{cos^{m-2} ax} qquadmbox{(for }m eq 1mbox{)},!

: intfrac{cos ax;dx}{sin^n ax} = -frac{1}{a(n-1)sin^{n-1} ax} +Cqquadmbox{(for }n eq 1mbox{)},!

: intfrac{cos^2 ax;dx}{sin ax} = frac{1}{a}left(cos ax+lnleft| anfrac{ax}{2} ight| ight) +C

: intfrac{cos^2 ax;dx}{sin^n ax} = -frac{1}{n-1}left(frac{cos ax}{asin^{n-1} ax)}+intfrac{dx}{sin^{n-2} ax} ight) qquadmbox{(for }n eq 1mbox{)}

: intfrac{cos^n ax;dx}{sin^m ax} = -frac{cos^{n+1} ax}{a(m-1)sin^{m-1} ax} - frac{n-m-2}{m-1}intfrac{cos^n ax;dx}{sin^{m-2} ax} qquadmbox{(for }m eq 1mbox{)},!

: also: intfrac{cos^n ax;dx}{sin^m ax} = frac{cos^{n-1} ax}{a(n-m)sin^{m-1} ax} + frac{n-1}{n-m}intfrac{cos^{n-2} ax;dx}{sin^m ax} qquadmbox{(for }m eq nmbox{)},!

: also: intfrac{cos^n ax;dx}{sin^m ax} = -frac{cos^{n-1} ax}{a(m-1)sin^{m-1} ax} - frac{n-1}{m-1}intfrac{cos^{n-2} ax;dx}{sin^{m-2} ax} qquadmbox{(for }m eq 1mbox{)},!

Integrals containing both sine and tangent

: int sin ax an ax;dx = frac{1}{a}(ln|sec ax + an ax| - sin ax)+C,!

: intfrac{ an^n ax;dx}{sin^2 ax} = frac{1}{a(n-1)} an^{n-1} (ax) +Cqquadmbox{(for }n eq 1mbox{)},!

Integrals containing both cosine and tangent

: intfrac{ an^n ax;dx}{cos^2 ax} = frac{1}{a(n+1)} an^{n+1} ax +Cqquadmbox{(for }n eq -1mbox{)},!

Integrals containing both sine and cotangent

: intfrac{cot^n ax;dx}{sin^2 ax} = frac{1}{a(n+1)}cot^{n+1} ax +Cqquadmbox{(for }n eq -1mbox{)},!

Integrals containing both cosine and cotangent

: intfrac{cot^n ax;dx}{cos^2 ax} = frac{1}{a(1-n)} an^{1-n} ax +Cqquadmbox{(for }n eq 1mbox{)},!

Integrals with symmetric limits

: int_-c^csin {x};dx = 0 !: int_-c^ccos {x};dx = 2int_0^ccos {x};dx = 2int_-c^0cos {x};dx = 2sin {c} !: int_-c^c an {x};dx = 0 !


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