NationMaster - Encyclopedia: List of integrals of trigonometric functions

The following is a list of integrals (antiderivative functions) of trigonometric functions. For a complete list of Integral functions, please see table of integrals and list of integrals. See also: trigonometric integral In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... Partial plot of a function f. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc functions... Trigonometric integrals are a family of integrals which involve trigonometric functions. ...

The constant c is assumed to be nonzero.

1 Integrals of trigonometric functions containing only sin
2 Integrals of trigonometric functions containing only cos
3 Integrals of trigonometric functions containing only tan
4 Integrals of trigonometric functions containing only sec
5 Integrals of trigonometric functions containing only csc
6 Integrals of trigonometric functions containing only cot
7 Integrals of trigonometric functions containing both sin and cos
8 Integrals of trigonometric functions containing both sin and tan
9 Integrals of trigonometric functions containing both cos and tan
10 Integrals of trigonometric functions containing both sin and cot
11 Integrals of trigonometric functions containing both cos and cot
12 Integrals of trigonometric functions containing both tan and cot

Integrals of trigonometric functions containing only sin

$intsin cx;dx = -frac{1}{c}cos cx,!$

$intsin^n {cx};dx = -frac{sin^{n-1} cxcos cx}{nc} + frac{n-1}{n}intsin^{n-2} cx;dx qquadmbox{(for }n>0mbox{)},!$

$intsqrt{1 - sin{x}},dx = intsqrt{operatorname{cvs}{x}},dx = 2 frac{cos{frac{x}{2}} + sin{frac{x}{2}}}{cos{frac{x}{2}} - sin{frac{x}{2}}} sqrt{operatorname{cvs}{x}}$

( = 2(1 + sin x)^0.5 ) In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

where cvs{x} is the Coversine function In trigonometry, the coversine, denoted cvs(x), of an angle is defined as one minus the sine of the angle: The derivative of the coversine is the negative of the cosine and the integral is Very few applications of this function exist, and it is generally only used to provide...

$int xsin cx;dx = frac{sin cx}{c^2}-frac{xcos cx}{c},!$

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

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

$intfrac{sin cx}{x} dx = sum_{i=0}^infty (-1)^ifrac{(cx)^{2i+1}}{(2i+1)cdot (2i+1)!},!$

$intfrac{sin cx}{x^n} dx = -frac{sin cx}{(n-1)x^{n-1}} + frac{c}{n-1}intfrac{cos cx}{x^{n-1}} dx,!$

$intfrac{dx}{sin cx} = frac{1}{c}ln left|tanfrac{cx}{2}right|$

$intfrac{dx}{sin^n cx} = frac{cos cx}{c(1-n) sin^{n-1} cx}+frac{n-2}{n-1}intfrac{dx}{sin^{n-2}cx} qquadmbox{(for }n>1mbox{)},!$

$intfrac{dx}{1pmsin cx} = frac{1}{c}tanleft(frac{cx}{2}mpfrac{pi}{4}right)$

$intfrac{x;dx}{1+sin cx} = frac{x}{c}tanleft(frac{cx}{2} - frac{pi}{4}right)+frac{2}{c^2}lnleft|cosleft(frac{cx}{2}-frac{pi}{4}right)right|$

$intfrac{x;dx}{1-sin cx} = frac{x}{c}cotleft(frac{pi}{4} - frac{cx}{2}right)+frac{2}{c^2}lnleft|sinleft(frac{pi}{4}-frac{cx}{2}right)right|$

$intfrac{sin cx;dx}{1pmsin cx} = pm x+frac{1}{c}tanleft(frac{pi}{4}mpfrac{cx}{2}right)$

$intsin c_1xsin c_2x;dx = frac{sin(c_1-c_2)x}{2(c_1-c_2)}-frac{sin(c_1+c_2)x}{2(c_1+c_2)} qquadmbox{(for }|c_1|neq|c_2|mbox{)},!$

Integrals of trigonometric functions containing only cos

$intcos cx;dx = frac{1}{c}sin cx,!$

$intcos^n cx;dx = frac{cos^{n-1} cxsin cx}{nc} + frac{n-1}{n}intcos^{n-2} cx;dx qquadmbox{(for }n>0mbox{)},!$

$int xcos cx;dx = frac{cos cx}{c^2} + frac{xsin cx}{c},!$

$int x^ncos cx;dx = frac{x^nsin cx}{c} - frac{n}{c}int x^{n-1}sin cx;dx,!$

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

$intfrac{cos cx}{x} dx = ln|cx|+sum_{i=1}^infty (-1)^ifrac{(cx)^{2i}}{2icdot(2i)!},!$

$intfrac{cos cx}{x^n} dx = -frac{cos cx}{(n-1)x^{n-1}}-frac{c}{n-1}intfrac{sin cx}{x^{n-1}} dx qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{dx}{cos cx} = frac{1}{c}lnleft|tanleft(frac{cx}{2}+frac{pi}{4}right)right|$

$intfrac{dx}{cos^n cx} = frac{sin cx}{c(n-1) cos^{n-1} cx} + frac{n-2}{n-1}intfrac{dx}{cos^{n-2} cx} qquadmbox{(for }n>1mbox{)},!$

$intfrac{dx}{1+cos cx} = frac{1}{c}tanfrac{cx}{2},!$

$intfrac{dx}{1-cos cx} = -frac{1}{c}cotfrac{cx}{2},!$

$intfrac{x;dx}{1+cos cx} = frac{x}{c}tanfrac{cx}{2} + frac{2}{c^2}lnleft|cosfrac{cx}{2}right|$

$intfrac{x;dx}{1-cos cx} = -frac{x}{c}cotfrac{cx}{2}+frac{2}{c^2}lnleft|sinfrac{cx}{2}right|$

$intfrac{cos cx;dx}{1+cos cx} = x - frac{1}{c}tanfrac{cx}{2},!$

$intfrac{cos cx;dx}{1-cos cx} = -x-frac{1}{c}cotfrac{cx}{2},!$

$intcos c_1xcos c_2x;dx = frac{sin(c_1-c_2)x}{2(c_1-c_2)}+frac{sin(c_1+c_2)x}{2(c_1+c_2)} qquadmbox{(for }|c_1|neq|c_2|mbox{)},!$

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Integrals of trigonometric functions containing only tan

$inttan cx;dx = -frac{1}{c}ln|cos cx|,!$

$inttan^n cx;dx = frac{1}{c(n-1)}tan^{n-1} cx-inttan^{n-2} cx;dx qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{dx}{tan cx + 1} = frac{x}{2} + frac{1}{2c}ln|sin cx + cos cx|,!$

$intfrac{dx}{tan cx - 1} = -frac{x}{2} + frac{1}{2c}ln|sin cx - cos cx|,!$

$intfrac{tan cx;dx}{tan cx + 1} = frac{x}{2} - frac{1}{2c}ln|sin cx + cos cx|,!$

$intfrac{tan cx;dx}{tan cx - 1} = frac{x}{2} + frac{1}{2c}ln|sin cx - cos cx|,!$

This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ...

Integrals of trigonometric functions containing only sec

$int sec{cx} , dx = frac{1}{c}ln{left| sec{cx} + tan{cx}right|}$

$int sec^n{cx} , dx = frac{sec^{n-1}{cx} sin {cx}}{c(n-1)} ,+, frac{n-2}{n-1}int sec^{n-2}{cx} , dx qquad mbox{ (for }n ne 1mbox{)},!$

$int frac{dx}{sec{x} + 1} = x - tan{frac{x}{2}}$

Secant is a term in mathematics. ...

Integrals of trigonometric functions containing only csc

$int csc{cx} , dx = -frac{1}{c}ln{left| csc{cx} + cot{cx}right|}$

$int csc^n{cx} , dx = -frac{csc^{n-1}{cx} cos{cx}}{c(n-1)} ,+, frac{n-2}{n-1}int csc^{n-2}{cx} , dx qquad mbox{ (for }n ne 1mbox{)},!$

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Integrals of trigonometric functions containing only cot

$intcot cx;dx = frac{1}{c}ln|sin cx|,!$

$intcot^n cx;dx = -frac{1}{c(n-1)}cot^{n-1} cx - intcot^{n-2} cx;dx qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{dx}{1 + cot cx} = intfrac{tan cx;dx}{tan cx+1},!$

$intfrac{dx}{1 - cot cx} = intfrac{tan cx;dx}{tan cx-1},!$

Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ...

Integrals of trigonometric functions containing both sin and cos

$intfrac{dx}{cos cxpmsin cx} = frac{1}{csqrt{2}}lnleft|tanleft(frac{cx}{2}pmfrac{pi}{8}right)right|$

$intfrac{dx}{(cos cxpmsin cx)^2} = frac{1}{2c}tanleft(cxmpfrac{pi}{4}right)$

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

$intfrac{cos cx;dx}{cos cx + sin cx} = frac{x}{2} + frac{1}{2c}lnleft|sin cx + cos cxright|$

$intfrac{cos cx;dx}{cos cx - sin cx} = frac{x}{2} - frac{1}{2c}lnleft|sin cx - cos cxright|$

$intfrac{sin cx;dx}{cos cx + sin cx} = frac{x}{2} - frac{1}{2c}lnleft|sin cx + cos cxright|$

$intfrac{sin cx;dx}{cos cx - sin cx} = -frac{x}{2} - frac{1}{2c}lnleft|sin cx - cos cxright|$

$intfrac{cos cx;dx}{sin cx(1+cos cx)} = -frac{1}{4c}tan^2frac{cx}{2}+frac{1}{2c}lnleft|tanfrac{cx}{2}right|$

$intfrac{cos cx;dx}{sin cx(1+-cos cx)} = -frac{1}{4c}cot^2frac{cx}{2}-frac{1}{2c}lnleft|tanfrac{cx}{2}right|$

$intfrac{sin cx;dx}{cos cx(1+sin cx)} = frac{1}{4c}cot^2left(frac{cx}{2}+frac{pi}{4}right)+frac{1}{2c}lnleft|tanleft(frac{cx}{2}+frac{pi}{4}right)right|$

$intfrac{sin cx;dx}{cos cx(1-sin cx)} = frac{1}{4c}tan^2left(frac{cx}{2}+frac{pi}{4}right)-frac{1}{2c}lnleft|tanleft(frac{cx}{2}+frac{pi}{4}right)right|$

$intsin cxcos cx;dx = frac{1}{2c}sin^2 cx,!$

$intsin c_1xcos c_2x;dx = -frac{cos(c_1+c_2)x}{2(c_1+c_2)}-frac{cos(c_1-c_2)x}{2(c_1-c_2)} qquadmbox{(for }|c_1|neq|c_2|mbox{)},!$

$intsin^n cxcos cx;dx = frac{1}{c(n+1)}sin^{n+1} cx qquadmbox{(for }nneq 1mbox{)},!$

$intsin cxcos^n cx;dx = -frac{1}{c(n+1)}cos^{n+1} cx qquadmbox{(for }nneq 1mbox{)},!$

$intsin^n cxcos^m cx;dx = -frac{sin^{n-1} cxcos^{m+1} cx}{c(n+m)}+frac{n-1}{n+m}intsin^{n-2} cxcos^m cx;dx qquadmbox{(for }m,n>0mbox{)},!$

also: $intsin^n cxcos^m cx;dx = frac{sin^{n+1} cxcos^{m-1} cx}{c(n+m)} + frac{m-1}{n+m}intsin^n cxcos^{m-2} cx;dx qquadmbox{(for }m,n>0mbox{)},!$

$intfrac{dx}{sin cxcos cx} = frac{1}{c}lnleft|tan cxright|$

$intfrac{dx}{sin cxcos^n cx} = frac{1}{c(n-1)cos^{n-1} cx}+intfrac{dx}{sin cxcos^{n-2} cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{dx}{sin^n cxcos cx} = -frac{1}{c(n-1)sin^{n-1} cx}+intfrac{dx}{sin^{n-2} cxcos cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{sin cx;dx}{cos^n cx} = frac{1}{c(n-1)cos^{n-1} cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{sin^2 cx;dx}{cos cx} = -frac{1}{c}sin cx+frac{1}{c}lnleft|tanleft(frac{pi}{4}+frac{cx}{2}right)right|$

$intfrac{sin^2 cx;dx}{cos^n cx} = frac{sin cx}{c(n-1)cos^{n-1}cx}-frac{1}{n-1}intfrac{dx}{cos^{n-2}cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{sin^n cx;dx}{cos cx} = -frac{sin^{n-1} cx}{c(n-1)} + intfrac{sin^{n-2} cx;dx}{cos cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{sin^n cx;dx}{cos^m cx} = frac{sin^{n+1} cx}{c(m-1)cos^{m-1} cx}-frac{n-m+2}{m-1}intfrac{sin^n cx;dx}{cos^{m-2} cx} qquadmbox{(for }mneq 1mbox{)},!$

also: $intfrac{sin^n cx;dx}{cos^m cx} = -frac{sin^{n-1} cx}{c(n-m)cos^{m-1} cx}+frac{n-1}{n-m}intfrac{sin^{n-2} cx;dx}{cos^m cx} qquadmbox{(for }mneq nmbox{)},!$

also: $intfrac{sin^n cx;dx}{cos^m cx} = frac{sin^{n-1} cx}{c(m-1)cos^{m-1} cx}-frac{n-1}{n-1}intfrac{sin^{n-1} cx;dx}{cos^{m-2} cx} qquadmbox{(for }mneq 1mbox{)},!$

$intfrac{cos cx;dx}{sin^n cx} = -frac{1}{c(n-1)sin^{n-1} cx} qquadmbox{(for }nneq 1mbox{)},!$

$intfrac{cos^2 cx;dx}{sin cx} = frac{1}{c}left(cos cx+lnleft|tanfrac{cx}{2}right|right)$

$intfrac{cos^2 cx;dx}{sin^n cx} = -frac{1}{n-1}left(frac{cos cx}{csin^{n-1} cx)}+intfrac{dx}{sin^{n-2} cx}right) qquadmbox{(for }nneq 1mbox{)}$

$intfrac{cos^n cx;dx}{sin^m cx} = -frac{cos^{n+1} cx}{c(m-1)sin^{m-1} cx} - frac{n-m-2}{m-1}intfrac{cos^n cx;dx}{sin^{m-2} cx} qquadmbox{(for }mneq 1mbox{)},!$

also: $intfrac{cos^n cx;dx}{sin^m cx} = frac{cos^{n-1} cx}{c(n-m)sin^{m-1} cx} + frac{n-1}{n-m}intfrac{cos^{n-2} cx;dx}{sin^m cx} qquadmbox{(for }mneq nmbox{)},!$

also: $intfrac{cos^n cx;dx}{sin^m cx} = -frac{cos^{n-1} cx}{c(m-1)sin^{m-1} cx} - frac{n-1}{m-1}intfrac{cos^{n-2} cx;dx}{sin^{m-2} cx} qquadmbox{(for }mneq 1mbox{)},!$

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Integrals of trigonometric functions containing both sin and tan

$int sin cx tan cx;dx = frac{1}{c}(ln|sec cx + tan cx| - sin cx),!$

$intfrac{tan^n cx;dx}{sin^2 cx} = frac{1}{c(n-1)}tan^{n-1} (cx) qquadmbox{(for }nneq 1mbox{)},!$

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ...

Integrals of trigonometric functions containing both cos and tan

$intfrac{tan^n cx;dx}{cos^2 cx} = frac{1}{c(n+1)}tan^{n+1} cx qquadmbox{(for }nneq -1mbox{)},!$

Integrals of trigonometric functions containing both sin and cot

$intfrac{cot^n cx;dx}{sin^2 cx} = frac{1}{c(n+1)}cot^{n+1} cx qquadmbox{(for }nneq -1mbox{)},!$

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ...

Integrals of trigonometric functions containing both cos and cot

$intfrac{cot^n cx;dx}{cos^2 cx} = frac{1}{c(1-n)}tan^{1-n} cx qquadmbox{(for }nneq 1mbox{)},!$

Integrals of trigonometric functions containing both tan and cot

$int frac{tan^m(cx)}{cot^n(cx)};dx = frac{1}{c(m+n-1)}tan^{m+n-1}(cx) - int frac{tan^{m-2}(cx)}{cot^n(cx)};dxqquadmbox{(for }m + n neq 1mbox{)},!$

Categories: Integrals | Trigonometry | Mathematics-related lists This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ... Trigonometry In trigonometry, the cotangent is a function (see trigonometric function) defined as: or An interpretation of the cotangent of an angle x is as follows. ...

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