§4.26 Integrals

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§4.26(i) Introduction
§4.26(ii) Indefinite Integrals
§4.26(iii) Definite Integrals
§4.26(iv) Inverse Trigonometric Functions
§4.26(v) Compendia

§4.26(i) Introduction

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Throughout this section the variables are assumed to be real. The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.

§4.26(ii) Indefinite Integrals

Notes:: These results may be verified by differentiation.
Keywords:: integrals, trigonometric functions
Referenced by:: ¶ ‣ §1.4(iv), §4.26(i)
Permalink:: http://dlmf.nist.gov/4.26.ii

4.26.1 $\int\mathop{\sin\/}\nolimits xdx=-\mathop{\cos\/}\nolimits x,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
A&S Ref:: 4.3.113
Permalink:: http://dlmf.nist.gov/4.26.E1
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4.26.2 $\int\mathop{\cos\/}\nolimits xdx=\mathop{\sin\/}\nolimits x.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
A&S Ref:: 4.3.114
Permalink:: http://dlmf.nist.gov/4.26.E2
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4.26.3 $\int\mathop{\tan\/}\nolimits xdx=-\mathop{\ln\/}\nolimits\!\left(\mathop{\cos\/}\nolimits x\right),$ $-\tfrac{1}{2}\pi<x<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\tan\/}\nolimits z$ : tangent function and $x$ : real variable
A&S Ref:: 4.3.115
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4.26.4 $\int\mathop{\csc\/}\nolimits xdx=\mathop{\ln\/}\nolimits\!\left(\mathop{\tan\/}\nolimits\tfrac{1}{2}x\right),$ $0<x<\pi$ .

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\tan\/}\nolimits z$ : tangent function and $x$ : real variable
A&S Ref:: 4.3.116
Permalink:: http://dlmf.nist.gov/4.26.E4
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4.26.5 $\int\mathop{\sec\/}\nolimits xdx=\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right),$ $-\frac{1}{2}\pi<x<\frac{1}{2}\pi$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)$ : inverse Gudermannian function, $\mathop{\sec\/}\nolimits z$ : secant function and $x$ : real variable
A&S Ref:: 4.3.117
Permalink:: http://dlmf.nist.gov/4.26.E5
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For the right-hand side see (4.23.41) and (4.23.42).

4.26.6 $\int\mathop{\cot\/}\nolimits xdx=\mathop{\ln\/}\nolimits\!\left(\mathop{\sin\/}\nolimits x\right),$ $0<x<\pi$ .

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
A&S Ref:: 4.3.118
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4.26.7 $\int e^{{ax}}\mathop{\sin\/}\nolimits\!\left(bx\right)dx=\frac{e^{{ax}}}{a^{2}+b^{2}}(a\mathop{\sin\/}\nolimits\!\left(bx\right)-b\mathop{\cos\/}\nolimits\!\left(bx\right)),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $a$ : real or complex constant and $x$ : real variable
A&S Ref:: 4.3.136
Permalink:: http://dlmf.nist.gov/4.26.E7
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4.26.8 $\int e^{{ax}}\mathop{\cos\/}\nolimits\!\left(bx\right)dx=\frac{e^{{ax}}}{a^{2}+b^{2}}(a\mathop{\cos\/}\nolimits\!\left(bx\right)+b\mathop{\sin\/}\nolimits\!\left(bx\right)).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $a$ : real or complex constant and $x$ : real variable
A&S Ref:: 4.3.137
Permalink:: http://dlmf.nist.gov/4.26.E8
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§4.26(iii) Definite Integrals

Notes:: For (4.26.12) and (4.26.13) see Copson (1935, pp. 137 and 227).
Keywords:: integrals, trigonometric functions
Permalink:: http://dlmf.nist.gov/4.26.iii

Throughout this subsection $m$ and $n$ are integers.

¶ Orthogonality Properties

Keywords:: trigonometric functions

4.26.9 $\int _{0}^{\pi}\mathop{\sin\/}\nolimits\!\left(mt\right)\mathop{\sin\/}\nolimits\!\left(nt\right)dt=0,$ $m\neq n$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : integer and $n$ : integer
A&S Ref:: 4.3.140
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4.26.10 $\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(mt\right)\mathop{\cos\/}\nolimits\!\left(nt\right)dt=0,$ $m\neq n$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $m$ : integer and $n$ : integer
A&S Ref:: 4.3.140
Permalink:: http://dlmf.nist.gov/4.26.E10
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4.26.11 $\int _{0}^{\pi}{\mathop{\sin\/}\nolimits^{{2}}}\!\left(nt\right)dt=\int _{0}^{\pi}{\mathop{\cos\/}\nolimits^{{2}}}\!\left(nt\right)dt=\tfrac{1}{2}\pi,$ $n\neq 0$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : integer
A&S Ref:: 4.3.141
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4.26.12 $\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(mt\right)}{t}dt=\begin{cases}\frac{1}{2}\pi,&m>0,\\ 0,&m=0,\\ -\frac{1}{2}\pi,&m<0.\end{cases}$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $m$ : integer
A&S Ref:: 4.3.142
Referenced by:: §4.26(iii)
Permalink:: http://dlmf.nist.gov/4.26.E12
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4.26.13 $\int _{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(t^{2}\right)dt=\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(t^{2}\right)dt=\frac{1}{2}\sqrt{\frac{\pi}{2}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.144
Referenced by:: §4.26(iii)
Permalink:: http://dlmf.nist.gov/4.26.E13
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§4.26(iv) Inverse Trigonometric Functions

Notes:: These results may be verified by differentiation.
Keywords:: integrals, inverse trigonometric functions
Referenced by:: ¶ ‣ §1.4(iv), §4.26(i)
Permalink:: http://dlmf.nist.gov/4.26.iv

4.26.14 $\int\mathop{\mathrm{arcsin}\/}\nolimits xdx=x\mathop{\mathrm{arcsin}\/}\nolimits x+(1-x^{2})^{{1/2}},$ $-1<x<1$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $x$ : real variable
A&S Ref:: 4.4.58 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E14
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4.26.15 $\int\mathop{\mathrm{arccos}\/}\nolimits xdx=x\mathop{\mathrm{arccos}\/}\nolimits x-(1-x^{2})^{{1/2}},$ $-1<x<1$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $x$ : real variable
A&S Ref:: 4.4.59 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E15
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4.26.16 $\int\mathop{\mathrm{arctan}\/}\nolimits xdx=x\mathop{\mathrm{arctan}\/}\nolimits x-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $-\infty<x<\infty$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.4.60 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E16
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4.26.17 $\int\mathop{\mathrm{arccsc}\/}\nolimits xdx=x\mathop{\mathrm{arccsc}\/}\nolimits x+\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{{1/2}}\right),$ $1<x<\infty$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.4.61 (is stated differently.)
Permalink:: http://dlmf.nist.gov/4.26.E17
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4.26.18 $\int\mathop{\mathrm{arcsec}\/}\nolimits xdx=x\mathop{\mathrm{arcsec}\/}\nolimits x-\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{{1/2}}\right),$ $1<x<\infty$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.4.62 (is stated differently.)
Permalink:: http://dlmf.nist.gov/4.26.E18
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4.26.19 $\int\mathop{\mathrm{arccot}\/}\nolimits xdx=x\mathop{\mathrm{arccot}\/}\nolimits x+\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $0<x<\infty$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.4.63 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E19
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4.26.20 $\int x\mathop{\mathrm{arcsin}\/}\nolimits xdx=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\mathop{\mathrm{arcsin}\/}\nolimits x+\frac{x}{4}(1-x^{2})^{{1/2}},$ $-1<x<1$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $x$ : real variable
A&S Ref:: 4.4.64 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E20
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4.26.21 $\int x\mathop{\mathrm{arccos}\/}\nolimits xdx=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\mathop{\mathrm{arccos}\/}\nolimits x-\frac{x}{4}(1-x^{2})^{{1/2}},$ $-1<x<1$ .

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $x$ : real variable
A&S Ref:: 4.4.66 (modified)
Permalink:: http://dlmf.nist.gov/4.26.E21
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§4.26(v) Compendia

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Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).